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Nonlocally related partial differential equation systems, the nonclassical method and applications Yang, Zhengzheng 2013

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Nonlocally Related Partial DifferentialEquation Systems, the Nonclassical Methodand ApplicationsbyZhengzheng YangB.Sc., Shandong University of Science and Technology, 2006M.Sc., Chinese Academy of Sciences, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September 2013c? Zhengzheng Yang 2013AbstractSymmetry methods are important in the analysis of differential equation (DE) sys-tems. In this thesis, we focus on two significant topics in symmetry analysis: non-locally related partial differential equation (PDE) systems and the application ofthe nonclassical method.In particular, we introduce a new systematic symmetry-based method for con-structing nonlocally related PDE systems (inverse potential systems). It is shownthat each point symmetry of a given PDE system systematically yields a nonlocallyrelated PDE system. Examples include applications to nonlinear reaction-diffusionequations, nonlinear diffusion equations and nonlinear wave equations. Moreover,it turns out that from these example PDEs, one can obtain nonlocal symmetries (in-cluding some previously unknown nonlocal symmetries) from some correspondingconstructed inverse potential systems.In addition, we present new results on the correspondence between two poten-tial systems arising from two nontrivial and linearly independent conservation laws(CLs) and the relationships between local symmetries of a PDE system and thoseof its potential systems.We apply the nonclassical method to obtain new exact solutions of the nonlin-ear Kompaneets (NLK) equationut = x?2(x4(?ux + ?u + ?u2))x,where ? > 0, ? ? 0 and ? > 0 are arbitrary constants. New time-dependent exactsolutions for the NLK equationut = x?2(x4(?ux + ?u2))x,for arbitrary constants ? > 0, ? > 0 are obtained. Each of these solutions isexpressed in terms of elementary functions. We also consider the behaviours ofthese new solutions for initial conditions of physical interest. More specifically,three of these families of solutions exhibit quiescent behaviour and the other twofamilies of solutions exhibit blow-up behaviour in finite time. Consequently, itturns out that the corresponding nontrivial stationary solutions are unstable.iiPrefaceChapter 4 is based on joint work with my supervisor George Bluman. The de-velopment of the symmetry-based method is a result of close collaboration withhim. Theorem 4.2.1 and Corollary 4.2.4 were worked out jointly. In addition, Iwas responsible for the constructions of the inverse potential systems listed in thisthesis, the classifications of point symmetries of such inverse potential systemsand the proof of Proposition 4.4.4. A version of Chapter 4 has been submitted forpublication.Chapter 5 is based on joint work with George Bluman and Shou-fu Tian. Aversion of Chapter 5 is published [36]. I was responsible for the computationof ?nonclassical symmetries? and the new stationary solutions F5 and F6 of thenonlinear Kompaneets (NLK) equation. I wrote the first draft of the ?nonclassicalanalysis? part of the manuscript.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Symmetries, Conservation Laws and Applications . . . . . . . . . . 62.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Symmetries of DE systems . . . . . . . . . . . . . . . . . . . . . 62.2.1 Lie groups and local groups of transformations . . . . . . 62.2.2 Lie algebra and Lie bracket . . . . . . . . . . . . . . . . 102.2.3 Jet spaces and prolongations . . . . . . . . . . . . . . . . 122.2.4 Infinitesimal methods for symmetries . . . . . . . . . . . 152.2.5 Contact and higher-order symmetries . . . . . . . . . . . 222.2.6 Equivalence transformations and symmetry classification 262.3 Conservation laws and the direct method . . . . . . . . . . . . . 282.3.1 Conservation laws . . . . . . . . . . . . . . . . . . . . . 282.3.2 The direct method . . . . . . . . . . . . . . . . . . . . . 303 Nonlocally Related PDE Systems and Applications . . . . . . . . . 353.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 CL-based method for constructing nonlocally related PDE systemsin 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.1 Potential systems and subsystems . . . . . . . . . . . . . 373.2.2 Subsystems . . . . . . . . . . . . . . . . . . . . . . . . . 41ivTable of Contents3.2.3 Procedure for constructing a tree of nonlocally related PDEsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Nonlocal symmetries . . . . . . . . . . . . . . . . . . . . . . . . 473.4 Nonlocally related systems in three or more dimensions . . . . . 533.5 Relationships between local symmetries of PDE systems . . . . . 563.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624 Symmetry-based Method for Constructing Nonlocally Related PDESystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 Nonlocally related PDE systems arising from point symmetries . 644.3 Examples of inverse potential systems arising from point symme-tries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.3.1 Nonlinear reaction-diffusion equation . . . . . . . . . . . 684.3.2 Nonlinear diffusion equation . . . . . . . . . . . . . . . . 754.3.3 Nonlinear wave equation . . . . . . . . . . . . . . . . . 794.4 Examples of nonlocal symmetries arising from the symmetry-basedmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.4.1 Nonlocal symmetries of nonlinear diffusion equation . . . 824.4.2 Nonlocal symmetries of nonlinear wave equation . . . . . 854.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885 New Exact Nonclassical Solutions of the NLK Equation . . . . . . . 895.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2 Lie?s classical method . . . . . . . . . . . . . . . . . . . . . . . 905.2.1 The invariant form method . . . . . . . . . . . . . . . . . 905.2.2 The direct substitution method . . . . . . . . . . . . . . 935.3 The nonclassical method . . . . . . . . . . . . . . . . . . . . . . 945.4 Nonclassical analysis of the NLK equation . . . . . . . . . . . . 955.4.1 Invariant solutions of the NLK equation . . . . . . . . . . 965.4.2 Nonclassical symmetries of the NLK equation . . . . . . 965.4.3 New exact solutions of the NLK equation . . . . . . . . . 995.4.4 Stationary solutions . . . . . . . . . . . . . . . . . . . . 1045.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 Concluding Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.2.1 To determine whether two PDE systems are nonlocally re-lated . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109vTable of Contents6.2.2 The existence of subsystems . . . . . . . . . . . . . . . . 1106.2.3 The relationship of symmetries of a given PDE system andthose of its potential systems . . . . . . . . . . . . . . . 1106.2.4 The application of the obtained nonlocal symmetries . . . 1116.2.5 Nonlocal symmetries for PDE systems with three or moreindependent variables . . . . . . . . . . . . . . . . . . . 111Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112viList of Tables3.1 Point symmetry classification for the nonlinear diffusion equation(3.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2 Point symmetry classification for the potential system (3.5) . . . . 503.3 Point symmetry classification for the potential system (3.19) . . . 524.1 Point symmetry classification for the reaction-diffusion equation(4.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Point symmetry classification for the PDE (4.49) . . . . . . . . . 834.3 Point symmetry classification for the PDE (4.63) . . . . . . . . . 844.4 Point symmetry classification for the nonlinear wave equation (4.64) 854.5 Point symmetry classification for the PDE (4.68) . . . . . . . . . 86viiList of Figures2.1 The action of exp(?X) and exp(? ?X). . . . . . . . . . . . . . . . . 253.1 A tree of nonlocally related PDE systems for the nonlinear waveequation (3.24). . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.1 The constructed inverse potential systems for the nonlinear reaction-diffusion equation (4.1) (Q(u) is arbitrary), with the arrows point-ing to the inverse potential systems. . . . . . . . . . . . . . . . . 714.2 The constructed inverse potential systems for the nonlinear reaction-diffusion equation (4.1) (Q(u) = u3), with the arrows pointing tothe inverse potential systems. . . . . . . . . . . . . . . . . . . . . 724.3 The constructed inverse potential systems for the nonlinear reaction-diffusion equation (4.1) (Q(u) = eu), with the arrows pointing tothe inverse potential systems. . . . . . . . . . . . . . . . . . . . . 734.4 The constructed inverse potential systems for the nonlinear reaction-diffusion equation (4.1) (Q(u) = u ln u), with the arrows pointingto the inverse potential systems. . . . . . . . . . . . . . . . . . . 754.5 The constructed inverse potential system for the nonlinear diffusionequation (4.45), with the arrows pointing to the inverse potentialsystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.1 (a) U(x) = b(x?c)x2, 0 < b < 1, c ? 0, x > 0. In (b), u(x, t) is given by(5.57) for x > 0, t > 0, with the arrow pointing in the direction ofincreasing t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.2 (a) U(x) = b(x?c)x2, 0 < b < 1, c > 0, x ? c. In (b), u(x, t) is given by(5.58) for x ? c, t > 0, with the arrow pointing in the direction ofincreasing t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.3 (a) U(x) = b(x?c)x2, b < 0, c > 0, 0 < x ? c. In (b), u(x, t) is given by(5.59) for 0 < x ? c, t > 0, with the arrow pointing in the directionof increasing t. . . . . . . . . . . . . . . . . . . . . . . . . . . . 103viiiList of Figures5.4 (a) U(x) = b(x?c)x2, b > 1, c > 0, x ? c. In (b), u(x, t) is given by(5.60) for 0 < x ? c, 0 < t < ?t0, with the arrow pointing in thedirection of increasing t. . . . . . . . . . . . . . . . . . . . . . . 1035.5 (a) U(x) = b(x?c)x2, b > 1, c ? 0, x > 0. In (b), u(x, t) is given by(5.61) for 0 < x ? c, 0 < t < ?t0, with the arrow pointing in thedirection of increasing t. . . . . . . . . . . . . . . . . . . . . . . 1045.6 The stationary solution V(x) = x?cx2, in (a) c > 0, in (b) c ? 0. . . . 1055.7 The stationary solution V(x) = x+a tan(ax )x2, in (a) x > 2api , in (b)2a(2k+1)pi < x ? xk. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.8 The stationary solution x?a tanh(ax )x2, x ? ?. . . . . . . . . . . . . . . 106ixAcknowledgementsI thank my supervisors, Dr. George Bluman and Dr. Alexei Cheviakov, for theirguidance, patience and encouragement throughout my PhD study.I thank my parents, my wife and my sister, for their support, understanding andencouragement throughout my life.xChapter 1IntroductionThe use of symmetries to investigate the solvability of equations can be tracedto the middle of the nineteenth century when Galois established the relationshipbetween the solvability of polynomial equations by radicals and their symmetrygroups. Motivated by Galois? work, Sophus Lie developed the theory of continuousgroups, i.e., Lie groups, to deal with the solvability of differential equations (DEs)by quadratures [66, 67].A symmetry of a DE system is a transformation which maps the solutions ofthe DE system to other solutions. In this thesis, our interest is limited to symme-tries that are connected local Lie groups (continuous symmetries), which can becharacterized by their infinitesimal generators. Throughout this thesis, ?symme-try? means ?continuous symmetry?. One important type of symmetry is a localsymmetry. Local symmetries include:? point symmetries: in evolutionary form, the components of an infinitesimalgenerator for dependent variables depend at most linearly on the first deriva-tives of dependent variables;? contact symmetries (exist only for scalar DEs): in evolutionary form, thecomponent of an infinitesimal generator for the dependent variable dependsat most on first derivatives of the dependent variable;? higher-order symmetries: in evolutionary form, the components of an in-finitesimal generator for dependent variables depend at most on finite orderderivatives of dependent variables.An important feature of a point symmetry is that one is able to find such asymmetry systematically by Lie?s algorithm. Lie?s algorithm for finding the pointsymmetries of a DE system is presented in [21, 25, 28, 29, 39, 53, 75, 76, 80, 87].In finding the point symmetries of a DE system, one need only find the compo-nents of their infinitesimal generators. The invariance conditions yield a systemof linear determining equations, which can be solved explicitly through variousexisting software packages. There are some popular programs for solving largeover-determined DE systems, e.g., DIFFGROB2 [69, 70], standard form [84], rif1Chapter 1. Introduction[85], CRACK [94], etc. Lie?s algorithm can be extended to find contact or higher-order symmetries, in which one needs to take the differential consequences of thegiven DE system into consideration.Once one obtains a symmetry of a DE system, various applications arise.? One can construct new solutions from known solutions.? One can reduce the order of a given ordinary differential equation (ODE).Moreover, one can obtain the solutions of the given ODE from those of thereduced ODE.? One can construct invariant solutions for a given partial differential equation(PDE) system.From the knowledge of the contact (point) symmetries, one is able to? determine whether a given scalar PDE (PDE system) can be invertibly mappedinto a linear scalar PDE (PDE system), and find such a mapping when it ex-ists [25, 29, 31, 62];? determine whether a linear PDE with variable coefficients can be invertiblymapped into a linear PDE with constant coefficients, and find such a map-ping when it exists [18, 25, 29, 30].In 1918, for a variational system, Emmy Noether introduced a method for con-structing conservation laws (CLs) from point symmetries of its action functional[73]. In 1921, Bessel-Hagen extended Noether?s work to include divergence sym-metries, which leave invariant the action functional to within a divergence term[14]. In [9], Anco and Bluman introduced a systematic procedure (direct method)to construct CLs for a given DE system. The direct method includes and extendsNoether?s theorem, since it can be applied to any DE system. Moreover, the directmethod is coordinate-independent.Topologically, continuous symmetries are not limited to local symmetries [2,25, 29, 39, 61, 75, 91, 92]. A symmetry that is not a local symmetry is called anonlocal symmetry. A special kind of nonlocal symmetry is a symmetry whose in-finitesimal generator depends on the integrals of the dependent variables. However,it is hard to find such a special kind of nonlocal symmetry of a given PDE systemby applying Lie?s algorithm directly to it. A way to seek nonlocal symmetries ofa given PDE system is through application of Lie?s algorithm to a nonlocally re-lated PDE system of the given PDE system. Two PDE systems are equivalent andnonlocally related if they have the following properties:2Chapter 1. Introduction(1) Any solution of either PDE system yields a solution of the other PDE system.(2) The solutions of either PDE system yield all solutions of the other PDE sys-tem.(3) The correspondence between the solutions of these two PDE systems is notone-to-one.Throughout this thesis, ?nonlocally related? means ?equivalent and nonlocally re-lated?.Nonlocally related PDE systems play a crucial rule in the nonlocal analysis ofa given PDE system, since one could extend local analysis methods to nonlocalones by applying local analysis methods to nonlocally related PDE systems. Priorto the new work presented in this thesis, there are two known systematic methodsto construct nonlocally related PDE systems [7, 19, 23?26, 29, 32, 43].(1) The CL-based method: Use k nontrivial local CLs of a given PDE system toconstruct a k-plet potential system of the given PDE system.(2) Exclude some dependent variables from a given PDE system to constructsubsystems.Since one is able to find local CLs, provided they exist, of a given PDE systemsystematically through the direct method, it is straightforward for one to constructpotential systems of the given PDE system systematically. To obtain subsystems,one can exclude some dependent variables by cross-differentiation or direct sub-stitution from the given PDE system. It is important to remark that all potentialsystems arising from nontrivial CLs of a given PDE system are nonlocally relatedto the given PDE system. However, not all subsystems are nonlocally related to thegiven PDE system.In the framework of nonlocally related PDE systems, nonlocal symmetries ofa given PDE system R{x, t; u} can arise from point symmetries of any PDE systemin a tree of nonlocally related PDE systems that includes R{x, t; u}. When suchnonlocal symmetries can be found for a given PDE system, one can use such sym-metries systematically to possibly generate further exact solutions from its knownsolutions, to construct new invariant solutions, to find nonlocal linearizations, or tofind additional nonlocal CLs.Finding exact solutions is an essential topic in the field of DEs. One can usesymmetries to find invariant solutions of a given DE system. Lie?s classical methodis based on the construction of invariants for a symmetry of a given DE system. Ifa given PDE system admits a symmetry group, then the invariant solutions cor-responding to this symmetry can be obtained by solving a reduced PDE system3Chapter 1. Introductionwith fewer independent variables than the given PDE system. In [15], (also see[27]), Lie?s classical method for finding invariant solutions was generalized to thenonclassical method. In the nonclassical method, invariant solutions arise from?nonclassical symmetries?, which keep only some subsets of the solutions invari-ant. By construction, the nonclassical method includes Lie?s classical method andthe direct method introduced by Clarkson and Kruskal in [45].In this thesis, we focus on nonlocally related PDE systems, the nonclassicalmethod and their applications. In particular, the following new results are obtained.? We present a relationship between two potential systems arising from twonontrivial and linearly independent local CLs of a given PDE system.? We find a correspondence between local symmetries of a given PDE systemand those of its potential systems.? We introduce a new systematic symmetry-based method for constructing non-locally related PDE systems and show that such nonlocally related PDEsystems yield nonlocal symmetries for specific examples. Moreover, somenonlocal symmetries are previously unknown.? We apply the nonclassical method to obtain new exact solutions of the di-mensional nonlinear Kompaneets (NLK) equation [60] given byut = x?2(x4(?ux + ?u + ?u2))x, (1.1)where ? > 0, ? ? 0 and ? > 0 are arbitrary constants. These new exactsolutions are expressible in terms of elementary functions and allow one tostudy stability properties with respect to initial data.In Chapter 2, we give a brief introduction to symmetries, CLs and their appli-cations.In Chapter 3, we present the known framework for constructing nonlocallyrelated PDE systems and their applications. Two different cases are discussed:PDE systems with two independent variables and PDE systems with three or moreindependent variables. We present the known CL-based method for constructingnonlocally related PDE systems (potential systems) for these two cases. We alsostate the method for constructing subsystems of a given PDE system. In addition,we present an extended procedure for constructing a tree of nonlocally related PDEsystems. Various examples are shown in this chapter. Moreover, we illustrate howto use nonlocally related PDE systems to find nonlocal symmetries and nonlocalCLs of a given PDE system. Two new results are presented. In particular, weshow that for two potential systems written in Cauchy-Kovalevskaya form, arising4Chapter 1. Introductionfrom two nontrivial and linearly independent local CLs of a given PDE system,the potential variable of one system cannot be expressed as a local function interms of the independent variables, dependent variables and their derivatives of theother system. Furthermore, we investigate relationships between symmetries ofsubsystems and those of potential systems. We prove that any local symmetry ofa PDE system with precisely n local CLs can be obtained by projection of somelocal symmetry of its n-plet potential system.In Chapter 4, we present a new systematic symmetry-based method for con-structing nonlocally related PDE systems. It is shown that any point symmetryof a given PDE system yields a nonlocally related PDE system (inverse potentialsystem). It turns out that the nonlocally related PDE systems arising from pointsymmetries can also yield nonlocal symmetries of a given PDE system. Someexamples are listed to illustrate this new method.In Chapter 5, firstly, we review Lie?s classical method for constructing invariantsolutions of a given PDE system. Following this, we give an introduction to thenonclassical method. Finally, we use the nonclassical method to construct exactsolutions of the NLK equation. It is shown that the nonclassical method can yieldfurther exact solutions beyond those arising from point symmetries of the NLKequation. Moreover, the new solutions are shown to be expressible in terms ofelementary functions. The properties of such new solutions are exhibited. It turnsout that these new solutions yield five families of solutions with initial conditions ofphysical interest. In particular, three of these families of solutions exhibit quiescentbehaviour, i.e., limt??u(x, t) = 0, and the other two families of solutions exhibit blowup behaviour, i.e., limt?t?u(x, t?) = ? for some finite t? depending on a constant intheir initial conditions. Moreover, new stationary solutions are presented.In Chapter 6, conclusions and some open problems are proposed.Throughout the thesis, we use the software package GeM to do necessary com-putations [40].5Chapter 2Symmetries, Conservation Lawsand Applications2.1 IntroductionIn this chapter we review the basic ideas of local symmetries and CLs. Lie?s algo-rithm for finding local symmetries of a DE system is discussed. We also presentequivalence transformations and symmetry classifications for a class of PDEs. Asan end of this chapter, we state the direct method for constructing CLs and someconnections between symmetries and CLs.2.2 Symmetries of DE systems2.2.1 Lie groups and local groups of transformationsIn applications, symmetries of a DE system are often Lie groups of transformationsacting on the solution manifold of the given DE.Definition 2.2.1 An r-parameter Lie group is an r-dimensional smooth manifoldG that is also a group with the property that the multiplication mapm : G ? G ? G, m(g, h) = g ? h, g, h ? G,and the inversioni : G ? G, i(g) = g?1, g ? G,are smooth.One significant application of Lie groups involves actions by Lie groups onspecial manifolds.Definition 2.2.2 A transformation group acting on a smooth manifold M is deter-mined by a Lie group G and a smooth map ?: G ? M ? M, denoted by ?(g, x) =g?x, which satisfies62.2. Symmetries of DE systems(1) e ? x = x, where e is the identity of G and x ? M.(2) g ? (h ? x) = (g ? h) ? x for all x ? M, g, h ? G.In many cases, we are only interested in local group action, i.e., for a givenx ? M, g ? x is only defined for elements g that lie in a small neighborhood of theidentity e.Definition 2.2.3 A local group of transformations acting on a smooth manifold Mis determined by a Lie group G, an open subset D, with {e} ? M ? D ? {G} ? M,and a smooth map ?: D? M, denoted by ?(g, x) = g ? x, which satisfies(1) For all x ? M, e ? x = x.(2) If (h, x) ? D, (g, h ? x) ? D, and (g ? h, x) ? D, then g ? (h ? x) = (g ? h) ? x.(3) If (g, x) ? D, then (g?1, g ? x) ? D and g?1 ? (g ? x) = x.Among transformation groups, a one-parameter group of transformations is animportant kind that plays a significant role in various fields.Definition 2.2.4 A (smooth) local one-parameter group of transformations (alsocalled a local flow) acting on a smooth manifold M is a local group of transforma-tions ?: D? M, where D ? R ? M is the flow domain, with the properties:(1) ?(?,?(?, x)) = ?(? + ?, x), x ? M, for all ?, ? ? R such that both sides of theequation are defined.(2) ?(0, x) = x, x ? M.If D = R ? M, ? is called a global one-parameter group of transformations (or aglobal flow).In order to distinguish the point x ? M and the parameter ?, we use the notation?(x; ?) to denote a one-parameter group of transformations.Example 2.2.5 Consider the map ?: R ? R2 ? R given by?(?, x, y) = ?(x, y; ?) = (x cos ? ? y sin ?, x sin ? + y cos ?).Then ? is a global one-parameter group of transformations denoting the rotationgroup on a plane.72.2. Symmetries of DE systemsTheorem 2.2.6 If ?: D ? M is a smooth local one-parameter group of transfor-mations, for each x ? M, define a vector byX|x =dd???????=0?(x; ?). (2.1)Then the assignment x 7? X|x is a smooth vector field on M, which is called theinfinitesimal generator of ?.Proof. See [64] for the proof.Suppose the dimension of M is n. In local coordinates, by Taylor?s formula,for ? in a small neighborhood of 0,?(x; ?) = x + ??(x) + O(?2),where ? = (?1, . . . , ?n) is given by?i(x) = ?????????=0?i(x; ?), i = 1, . . . , n. (2.2)Thus the infinitesimal generator of ? is given byX =n?i=1?i(x) ??xi. (2.3)The quantities ?i, i = 1, . . . , n, are called the infinitesimals of the one-parametergroup of transformations ?. The transformationx? = x + ? ?(x) (2.4)defines the infinitesimal transformation of ?.Example 2.2.7 Consider the rotation group ? in Example 2.2.5. According toformula (2.3), the infinitesimal generator of ? is given byX = ?y ??x+ x ??y.On the other hand, the infinitesimal generators can be used to characterize aone-parameter group of transformations. One can obtain the one-parameter groupof transformations generated by a smooth vector X through solving an ODE sys-tem. We use the notation exp(?X) to denote the one-parameter group of transfor-mations generated by X, i.e.,exp(?X) = ?(x; ?).82.2. Symmetries of DE systemsOne can show thatexp(?X)x = x + ??(x) + ?22X(?)(x) + ? ? ? =??k=0?kk! Xk(x), (2.5)where X is given by (2.3), ? = (?1, . . . , ?n), X(?) = (X(?1), . . . ,X(?n)), and Xk =XXk?1.In local coordinates, the one-parameter group of transformations ?(x; ?) canbe determined from its infinitesimal generator through Lie?s First FundamentalTheorem.Theorem 2.2.8 (Lie?s First Fundamental Theorem) A one-parameter group oftransformations ?(x; ?) is equivalent to the solution of the initial value problem fora system of first order ODEsdd??(x; ?) = ? (?(x; ?)) , ?(x; 0) = x. (2.6)Proof. See [21, 29, 48, 75, 80] for the proof.Example 2.2.9 Consider the infinitesimal generatorX = 2t ??x? xu ??u. (2.7)Here the one-parameter group of transformations ?(?) = (?t(?), x?(?), u?(?)) gener-ated by X satisfiesd?td? = 0,dx?d? = 2?t, (2.8)du?d? = ?x?u?,with initial value ?(0) = (t, x, u). Solving the equations (2.8), one finds that theone-parameter group of transformations ? generated by X is given by?t = t,x? = x + 2?t, (2.9)u? = ue??x??2 t.The following shows that, by choosing proper local coordinates, a vector fieldnear a regular point x0, i.e., Xx0 , 0, can be expressed in a simple canonical form[21, 25, 29, 39, 53, 64, 75, 76, 80].92.2. Symmetries of DE systemsTheorem 2.2.10 Suppose X is a smooth vector field on a smooth manifold M, andXx0 , 0 at a point x0 ? M. Then there exist smooth coordinates (y1, . . . , yn) onsome neighborhood of x0 such that X has the coordinate representation (canonicalform) ??yn .If y = f (x) is a change of coordinates, then the vector fieldX =n?i=1?i(x) ??xi(2.10)has the expressionX =n?j=1n?i=1?i( f ?1(y))? fj?xi( f ?1(y)) ??y j(2.11)in the y coordinates. Suppose the corresponding canonical coordinates of the vectorfield X are given byy1 = f 1(x),y2 = f 2(x),? ? ?yn = f n(x).(2.12)Since the canonical form of X is given by ??yn , according to the formula (2.11),f (x) = ( f 1(x), . . . , f n(x)) satisfies the following first order linear PDE systemn?i=1?i(x)? f?(x)?xi= ?y??yn = 0, ? = 1, . . . , n ? 1,n?i=1?i(x)? fn(x)?xi= ?yn?yn = 1.(2.13)The canonical form of a vector field is essential in the symmetry-based methodfor constructing nonlocally related PDE systems, which will be presented in Chap-ter 4.2.2.2 Lie algebra and Lie bracketDefinition 2.2.11 A Lie algebra is a vector space g endowed with an operation[ , ]: g ? g? g, called the Lie bracket for g, that satisfies the following propertiesfor all X, Y, Z ? g:102.2. Symmetries of DE systems(1) Bilinearity: For a, b ? R,[aX + bY,Z] = a[X,Y] + b[X,Z],[X, aY + bZ] = a[X,Y] + b[X,Z].(2) Antisymmetry:[X,Y] = ?[Y,X].(3) Jacobi identity:[X, [Y,Z]] + [Z, [X,Y]] + [Y, [Z,X]] = 0.Definition 2.2.12 Let X and Y be two smooth vector fields on a smooth manifoldM. The Lie bracket of X and Y is a smooth vector field [X, Y]: C?(M) ? C?(M),where C?(M) denotes the set of all smooth real-valued functions on M, defined by[X,Y] = X(Y( f )) ? Y(X( f )) f ? C?(M). (2.14)An important property of the Lie bracket is given by the following theorem.Theorem 2.2.13 Suppose F: M ? N is a diffeomorphism and X, Y ? T (M),where T (M) denotes the set of all smooth vector fields on M. Then F?[X,Y] =[F?(X), F?(Y)], where F? is the pushforward associated with F.Proof. See [64] for the proof.Suppose g is an r-dimensional Lie algebra. Let {X1, . . . ,Xr} be a basis of g,then [Xi,X j] ? g, i.e., there are specific constants cki j, i, j, k = 1, . . . , r, called thestructure constants of g, such that[Xi,X j] =r?kcki jXk, i, j = 1, . . . , r. (2.15)The structure constants have the following properties.Theorem 2.2.14 (Lie?s Third Fundamental Theorem) The structure constantssatisfy(1) Antisymmetry:cki j = ?ckji, (2.16)(2) Jacobi identity:r?k=1(cki jcmkl + cklicmk j + ckjlcmki)= 0. (2.17)Proof. See [48] for the proof.112.2. Symmetries of DE systems2.2.3 Jet spaces and prolongationsThe basic objects we consider in this thesis are DE systems. In order to applyinfinitesimal methods to DE systems, it is necessary to extend the basic space ofindependent variables x = (x1, . . . , xn) and dependent variables u = (u1, . . . , um) toa space including the derivatives of u. Let X ? Rn denote the space of n indepen-dent variables and U ? Rm denote the space of m dependent variables.Consider a smooth function f : X ? U, i.e., u = f (x) = ( f 1(x), . . . , f m(x)).For each i, there arepk ?(n + k ? 1k)different k-th order partial derivatives of f i(x). Let?Jg(x) = ?kg(x)?x j1?x j2 ? ? ? ?x jkfor every smooth scalar-valued function g(x) = g(x1, . . . , xn), where J = ( j1, . . . , jk)is an unordered k-tuple of integers with 1 ? j? ? n for ? = 1, . . . , k. We call J anunordered multi-index of order k, i.e., |J | = k. We use the notation?k f (x) = (?k f 1(x), . . . , ?k f m(x))to denote the k-th order partial derivatives of f (x). In particular, ? f (x) = ?1 f (x).Hence, there are m ? pk different k-th order partial derivatives of f (x). It followsthat one needs m ? pk different coordinates uiJ , i = 1, . . . ,m, |J | = k to represent alldifferent k-th order partial derivativesuiJ = uij1 ,..., jk =?kui?x j1?x j2 ? ? ? ?x jk= ?J f i(x)of a function u = f (x). Let Uk ? Rm?pk be the space of all k-th order partialderivatives of u. Consequently, the space of all the derivatives of u up to l is U(l) ?U ?U1 ? ? ? ? ? Ul, whose dimension ism + mp1 + ? ? ? + mpl = m(n + ll)? mp(l).We denote a point in U(l) by u(l). For the smooth function u = f (x), the l-thprolongation of f , denoted by u(l) = f (l)(x), is defined by the equationsuiJ = ?J f i(x), i = 1, . . . ,m,where 0 ? |J | ? l. (By convention, ui0 denotes the component ui of u.)122.2. Symmetries of DE systemsDefinition 2.2.15 If M ? X ?U is an open set, the l-jet space of M is given byM(l) ? M ?U1 ? ? ? ? ? Ul.A (local) point transformation acting on M ? X ? U is defined by a (local)diffeomorphism on M:(x?, u?) = (F(x, u),G(x, u)) (x, u) ? M. (2.18)Let K ? X be an open set, and f : K ?U be a continuous function. The graphof f is given by?( f ) = {(x, u) ? Rn ? Rm : x ? K and u = f (x)} .The l-th order prolongation of a graph ?( f ) is given by?(l)( f ) ={(x, u, ?u, . . . , ?lu) : x ? K, (u, ?u, . . . , ?lu) = ( f (x), ? f (x), . . . , ?l f (x)}.If ? is a local point transformation on M:x? = ?1(x, u),u? = ?2(x, u), (2.19)and u = f (x) is a smooth function, then ? acts on u = f (x) by acting on its graph.Hence, it is natural to extend ? to a map ?(l) acting on the l-th jet space of M, whichmaps the derivatives of u = f (x) to the derivatives of the transformed functionu? = ?f (x?). The l-th order prolongation of ? is given by ?(l) which satisfies?(l)(?(l)( f ))= ?(l)( ?f ). (2.20)An l-th order (local) contact transformation is a (local) diffeomorphism of M(l)onto itself:x?i = Fi(x, u(l)), i = 1, . . . , n,u?jJ = GjJ(x, u(l)), j = 1, . . . ,m, |J | ? l,(2.21)for (x, u(l)) ? M(l) with the contact conditions:du? jJ ?n?i=1u?jJ,idx?i = du jJ ?n?i=1ujJ ,idxi = 0, |J | < l. (2.22)For example, the Legendre transformationx? = ux,u? = u ? xux,u?x? = ?x,is a first order contact transformation that is not a first order prolongation of a pointtransformation. The following significant theorem is due to Ba?cklund [12].132.2. Symmetries of DE systemsTheorem 2.2.16 If there is more than one dependent variable, m > 1, then everycontact transformation is the prolongation of a point transformation. If m = 1,there exist first order contact transformations that are not first order prolongationsof point transformations. However, every l-th order contact transformation is thel-th order prolongation of a first order contact transformation.Proof. See [12, 76] for the proof.Consider the infinitesimal generator X of a local one-parameter group of trans-formations exp(?X) on M ? X ? U. We define the l-th order prolongation of XbyX(l)???(x,u(l)) =dd???????=0exp(?X)(l)(x, u(l)), (2.23)In local coordinates, X(l) can be computed by an explicit formula.Definition 2.2.17 The total derivative with respect to xi is given by the differentialoperatorDi = Dxi =??xi+m?j=1?JujJ,i??u jJ, (2.24)where the summation over the multi-indices J is over all J?s with |J | ? 0.Theorem 2.2.18 LetX =n?i=1?i(x, u) ??xi+m?j=1? j(x, u) ??u jbe a smooth vector field on M. The l-th order prolongation of X is the smoothvector fieldX(l) = X +m?j=1?J? jJ(x, u(l))??u jJ(2.25)with the coefficients? jJ(x, u(l)) = DJ???????? j ?n?i=1?iu ji??????? +n?i=1?iu jJ,i, (2.26)where the summation in (2.25) over the multi-indices J is over all unordered multi-indices J = ( j1, . . . , jk) with 1 ? j? ? n for ? = 1, . . . , k, 1 ? k ? l, and DJ =D j1 D j2 ? ? ?D jk .Proof. See [21, 25, 29, 39, 53, 75, 76, 80] for the proof.Let X and Y be two smooth vector fields on M ? X ? U, then their prolonga-tions have the properties [75, 76]:142.2. Symmetries of DE systems(1) Linearity: For c1, c2 ? R,(c1X + c2Y)(l) = c1X(l) + c2Y(l).(2) Lie bracket property:[X,Y](l) =[X(l),Y(l)].2.2.4 Infinitesimal methods for symmetriesBefore discussing symmetries of a DE system, it is necessary to consider a sim-pler case: symmetries of a system of algebraic equations. Consider a system ofalgebraic equations defined for x in some manifold M:F?(x) = 0, ? = 1, . . . , s, (2.27)where F?(x) ? C?(M), ? = 1, . . . , s.Definition 2.2.19 Let M be a manifold and S ? M. A local group of transforma-tions G acting on M is a symmetry group of S if whenever x ? S , and g ? G is suchthat g ? x is defined, then g ? x ? S .Let S F = {x : F?(x) = 0, ? = 1, . . . , s} be the set of solutions of the algebraicequation system (2.27). A local group of transformations G is a symmetry group ofthe algebraic equation system (2.27) if it is a symmetry group of S F .Definition 2.2.20 Let M and N be two manifolds, and let G be a local group oftransformations acting on M. A function f : M ? N is a G-invariant function iff (g ? x) = f (x) for all x ? M and all g ? G such that g ? x is defined. If N = R, f iscalled an invariant of G.Theorem 2.2.21 Let G be a connected group of transformations acting on a mani-fold M. Then ? ? C?(M) is an invariant for G if and only if for every infinitesimalgenerator X of G,X(?) = 0, for all x ? M. (2.28)Proof. See [75, 76, 80] for the proof.Definition 2.2.22 Let M be a smooth manifold, ?1, . . . , ?k ? C?(M) are function-ally dependent if for arbitrary x0 ? M there exists a neighborhood Ux0 of x0 and afunction F(y1, . . . , yk) ? C?(Rk) with F . 0 on any open subset of Rk, such thatF(?1(x), . . . , ?k(x))= 0, (2.29)for all x ? Ux0 . Otherwise, ?1, . . . , ?k are functionally independent.152.2. Symmetries of DE systemsTheorem 2.2.23 Let exp(?X) be a one-parameter group of transformations act-ing on an n-dimensional smooth manifold M, and let x0 ? M be a regular pointfor X. Then exp(?X) has precisely m ? 1 functionally independent local invari-ants ?1(x), . . . , ?n?1(x) defined in a neighborhood of x0. Moreover, any other localinvariant of exp(?X) defined near x0 is of the form?(x) = F(?1(x), . . . , ?m?1(x)), (2.30)for some smooth function F.Proof. See [75, 76] for the proof.Theorems 2.2.21 and 2.2.23 provide a method for constructing invariants of aone-parameter group of transformations near a regular point x0. LetX =n?i=1?i(x) ??xibe the infinitesimal generator of a one-parameter group of transformations exp(?X).According to Theorem 2.2.21, a local invariant ?(x) of exp(?X) satisfiesX(?) =n?i=1?i(x) ???xi= 0. (2.31)Theorem 2.2.23 implies there exist n ? 1 functionally independent local invariantsof exp(?X) near x0. The general solution of the linear homogeneous first order PDE(2.31) can be obtained through solving the corresponding characteristic system ofODEsdx1?1(x) =dx2?2(x) = ? ? ? =dxn?n(x) . (2.32)If the general solution of (2.32) is given by?1(x1, . . . , xn) = c1, . . . , ?n?1(x1, . . . , xn) = cn?1,where ci?s are constants, then ?1, . . . , ?n?1 are functionally independent local in-variants.Example 2.2.24 Consider the one parameter rotation group SO(2) with the in-finitesimal generator X = ?y ??x + x??y . The solution for its corresponding charac-teristic systemdx?y =dyx(2.33)162.2. Symmetries of DE systemsis given by x2 + y2 = c. Thus, ? = x2 + y2 is a local invariant of SO(2), and anylocal invariant of SO(2) is a function of ? = x2 + y2. Geometrically, the distancebetween a point and the origin is invariant under the rotation group SO(2).Theorem 2.2.23 also assures one that there exist n?1 functionally independentsolutions ( f 1(x), . . . , f n?1(x)) of the PDE system (2.13) which are used to obtaincorresponding canonical coordinates of a vector field X. In order to obtain f n(x),one introduces a new variable v and solves the following linear homogeneous firstorder PDEn?i=1?i(x) ???xi+ ???v = 0, (2.34)using the method of characteristics. Sincen?i=1?i(x)? fn(x)?xi= 1if and only if ? = v ? f n(x) is a solution of (2.34), one can obtain f n(x) from thesolution of (2.34). Thus the n ? 1 functionally independent functions ( f 1(x), . . . ,f n?1(x)), together with the function f n(x), yield the corresponding canonical coor-dinates of X.Example 2.2.25 Consider the vector field X = ?y ??x+x ??y. Suppose the canonicalcoordinates of X are given byz = f (x, y),w = g(x, y). (2.35)From Example 2.2.24, one obtains f (x, y) = x2 + y2. To obtain g(x, y), one firstfinds the invariants of Y = X + ??v , i.e., solves the ODEsdx?y =dyx= dv1. (2.36)Since r =?x2 + y2 is an invariant of Y, one replaces x by?r2 ? y2 in (2.36). Thisyields the following equationdy?r2 ? y2= dv1. (2.37)Integrating (2.37) leads to another invariant of Y given by? = v ? arcsin yr= v ? arctan yx.172.2. Symmetries of DE systemsThus the canonical coordinates of X are given byz = x2 + y2,w = arctan yx,(2.38)and X = ??w in (z,w) coordinates.Theorem 2.2.26 Suppose the algebraic equation system (2.27) is of maximal rank,i.e., the Jacobian matrix(?F??xk)is of rank s for every x ? S F . Then G is a symmetrygroup of the algebraic equation system (2.27) if and only if for every infinitesimalgenerator X of G,X(F?(x)) = 0, ? = 1, . . . , s, (2.39)wheneverF?(x) = 0, ? = 1, . . . , s.Proof. See [75] for the proof.For the case of a DE system, consider a DE system R{x; u} of s DEs of orderl with n independent variables x = (x1, . . . , xn) and m dependent variables u =(u1, . . . , um) given byR?(x, u(l)) = R?(x, u, ?u, . . . , ?lu) = 0, ? = 1, . . . , s. (2.40)A solution of the DE system R{x; u} (2.40) is a smooth function u = f (x) satisfyingR?(x, f (l)(x)) = 0, ? = 1, . . . , s,when x is in the domain of f , i.e.,?(l)f ={(x, f (l)(x))}? S R{x;u} ={(x, u(l)) : R?(x, u(l)) = 0, ? = 1, . . . , s.}Definition 2.2.27 A point symmetry of the DE system R{x; u} (2.40) is a local one-parameter group of transformations G that leaves invariant the solution manifold ofR{x; u} (2.40), i.e., if u = f (x) is a solution of R{x; u} (2.40), and the transformedfunction ?f = g ? f is defined for g ? G, then u? = ?f (x?) is also a solution of R{x; u}(2.40).In order to apply the infinitesimal method to the DE system R{x; u} (2.40), it isnecessary to impose some additional conditions on R{x; u} (2.40).182.2. Symmetries of DE systemsDefinition 2.2.28 Consider the DE system R{x; u} (2.40). The system is of maxi-mal rank if the rank of its Jacobian matrixJ(x, u(l)) =?????????R??xi, ?R??u jJ????????s?(n+mp(l))(2.41)with respect to the variables (x, u(l)) is s whenever R?(x, u(l)) = 0, ? = 1, . . . , s.The maximal rank condition eliminates the redundancy of a DE system.Theorem 2.2.29 Suppose the DE system R{x; u} (2.40) is of maximal rank. Ifevery infinitesimal generator X of a local group of transformations G satisfiesX(l)R?(x, u(l)) = 0, ? = 1, . . . , s, (2.42)wheneverR?(x, u(l)) = 0, ? = 1, . . . , s,then G is a point symmetry of R{x; u} (2.40).Proof. See [21, 25, 29, 39, 53, 75, 76, 80] for the proof.Theorem 2.2.29 provides a systematic way to find point symmetry of a DE sys-tem with maximal rank. However, there is no assurance that all point symmetriesare found. In order to obtain a necessary and sufficient condition, the given DEsystem must satisfy another condition.Definition 2.2.30 Consider the DE system R{x; u} (2.40). The system is locallysolvable at the point (x0, u(l)0 ) ? S R{x;u} if there exists a smooth solution u = f (x),defined in a neighborhood of x0, which satisfies u(l)0 = f (l)(x0). A system is said tobe locally solvable if it is locally solvable at every point in S R{x;u}.Definition 2.2.31 A DE system is nondegenerate if at every point (x0, u(l)0 ) ? M(l) itis both of maximal rank and locally solvable. A DE system is totally nondegenerateif it and all its differential consequences are nondegenerate.Throughout the thesis, unless stated otherwise, all DE systems are assumed tobe totally nondegenerate.Theorem 2.2.32 Suppose the DE system R{x; u} (2.40) is nondegenerate. A localgroup of transformations G is a point symmetry of R{x; u} (2.40) if and only ifX(l)R?(x, u(l)) = 0, ? = 1, . . . , s, (2.43)wheneverR?(x, u(l)) = 0, ? = 1, . . . , s,for every infinitesimal generator X of G.192.2. Symmetries of DE systemsProof. See [21, 25, 29, 39, 53, 75, 76, 80] for the proof.According to the properties of prolonged vector fields, the set of all infinitesi-mal symmetries of a nondegenerate PDE system forms a Lie algebra.Definition 2.2.33 An l-th order DE system is regular if it is of maximal rank,analytic and contains all its differential consequences up to order l, i.e., no furtherdifferential consequences of order l or less can be obtained from the DE systemthrough differentiation or taking integrability conditions.Theorem 2.2.34 A regular DE system is nondegenerate.Proof. See [68] and references therein for the proof.Theorem 2.2.32 provides us a systematic way to find the point symmetries ofan l-th order nondegenerate DE system.Algorithm 2.2.35 (Lie?s algorithm for finding point symmetries): Consider anl-th order nondegenerate DE system R{x; u}.1. Let the infinitesimal generator X of the point symmetries be of the formX =n?i=1?i(x, u) ??xi+m?j=1? j(x, u) ??u j,where the infinitesimals ?i(x, u) and ? j(x, u) are unknown functions of x andu to be determined.2. Find the l-th order prolongation X(l) of X according to Theorem 2.2.32.3. Apply the l-th order prolongation X(l) to R{x; u}, and eliminate the depen-dencies among the derivatives of u arising from R{x; u} itself.4. Set the coefficients of the remaining derivatives of u to be zero. This stepyields a system of linear PDEs for the unknown functions ?i(x, u) and ? j(x, u),called the determining equations of the infinitesimals of the point symme-tries of R{x; u}.5. Solve the determining equations explicitly to obtain the general solutions of?i(x, u) and ? j(x, u).6. Exponentiate the infinitesimal generator X to obtain the global symmetrygroups.202.2. Symmetries of DE systemsExample 2.2.36 Consider the nonlinear reaction-diffusion equationut ? uxx = u3. (2.44)Since the nonlinear reaction-diffusion equation (2.44) is totally nondegenerate, onecan apply Algorithm 2.2.35 to find its all point symmetries. LetX = ?(x, t, u) ??x+ ?(x, t, u) ??t+ ?(x, t, u) ??u(2.45)be the infinitesimal generator of a point symmetry of the nonlinear reaction-diffusionequation (2.44). Then X is an infinitesimal point symmetry of the nonlinear reaction-diffusion equation (2.44) if and only if its second order prolongation X(2) satisfiesX(2)(ut ? uxx ? u3)???ut=uxx+u3= 0. (2.46)According to Step 4 in Algorithm 2.2.35, one obtains the following determiningequationsu3?uu ? ?uu + ?xu = 0,2?x ? u3?u + ?xx ? ?t = 0,2u3?xu ? ?t ? u3?u + ?xx ? 2?xu = 0,?t ? ?xx ? 3u2?u + u3?u ? u3?t ? u6?u + u3?xx = 0,?x = 0, ?u = 0, ?uu = 0, ?uu = 0, ?u + ?xu = 0.(2.47)Direct computation shows that the nonlinear reaction-diffusion equation (2.44)only has the three point symmetries given by the infinitesimal generatorsX1 =??x , X2 =??t ,X3 = x??x + 2t??t ? u??u .(2.48)Moreover, the corresponding one-parameter groups of transformations Gi gener-ated by Xi are given byG1 : (x?, ?t, u?) = (x + ?, t, u),G2 : (x?, ?t, u?) = (x, t + ?, u),G3 : (x?, ?t, u?) = (e?x, e2?t, e??u).(2.49)212.2. Symmetries of DE systems2.2.5 Contact and higher-order symmetriesThe infinitesimals for a point symmetry depend only on x and u. A natural exten-sion of the notion of point symmetry is by allowing the infinitesimals to dependon derivatives of u. We use the notation P[u] to denote P as a smooth functiondepending on x, u and derivatives of u.Definition 2.2.37 A generalized vector field is an expression of the formX =n?i=1?i[u] ??xi+m?j=1? j[u] ??u j, (2.50)where ?i and ? j are smooth functions.Analogous to the prolongation of a smooth vector field, the l-th prolongationof a generalized vector field X is given byX(l) = X +m?j=1?1?|J |?l? jJ[u]??u jJ(2.51)with the coefficients? jJ[u] = DJ???????? j ?n?i=1?iu ji??????? +n?i=1?iu jJ,i, (2.52)with the same notation as in Theorem 2.2.14. In particular, the infinite prolongationof (2.50) is the infinite sumX(?) = X +m?j=1?|J |?1? jJ[u]??u jJ, (2.53)where ? jJ[u] is given by (2.52).Theorem 2.2.38 If the number of dependent variables is one, i.e., m = 1, a gen-eralized vector field X is an infinitesimal generator of a one-parameter group of a(first order) contact transformation if and only if ?i[u] = ?i(x, u, ?u) and ?1[u] =?1(x, u, ?u) satisfy??1[u]?u j?n?i=1ui??i[u]?u j= 0, j = 1, . . . , n. (2.54)Proof. See [25, 76] for the proof.222.2. Symmetries of DE systemsDefinition 2.2.39 A generalized vector field X is a higher-order infinitesimal sym-metry (or an infinitesimal generator of a higher-order symmetry) of a DE systemR?[u] = 0, ? = 1, . . . , s, (2.55)if and only ifX(?)R?[u] = 0, ? = 1, . . . , s, (2.56)on any solution of (2.55). In particular, for m = 1, if X is an infinitesimal gen-erator of a one-parameter group of contact transformations, we call X a contactinfinitesimal symmetry.Definition 2.2.40 A local symmetry is a point symmetry, contact symmetry orhigher-order symmetry.In practice, it is useful to consider the generalized vector field with ?i[u] = 0.Definition 2.2.41 A generalized vector fieldXQ =m?j=1Q j[u] ??u jis called an evolutionary generalized vector field. The m-tuple of differential func-tions Q[u] = (Q1[u], . . . ,Qm[u]) is called its characteristic.The evolutionary form of a generalized vector field X (2.50) is given by?X =m?j=1???????? j ?n?i=1?iu ji?????????u j. (2.57)The m-tuple of differential functionsQ[u] = (Q1[u], . . . ,Qm[u]) =????????1 ?n?i=1?iu1i , . . . , ?m ?n?i=1?iumi??????? (2.58)is the corresponding characteristic of the generalized vector field X.Consider the one-parameter group of point transformations generated by theinfinitesimal generatorX =n?i=1?i(x, u) ??xi+m?j=1? j(x, u) ??u j. (2.59)232.2. Symmetries of DE systemsgiven byx? = exp(?X)x,u? = exp(?X)u, (2.60)with x = (x1, . . . , xn) and u = (u1, . . . , um). Let u = f (x) be a surface in X ? Uspace. The one-parameter group of point transformations exp(?X) maps u = f (x)into a family of surfaces u = g(x; ?) in X ?U space. According to the property ofone-parameter group of transformations, one obtainsx = exp(??X)x? = x? ? ??(x?, f (x?)) + O(?2),u = exp(??X)u? = u? ? ??(x?, f (x?)) + O(?2), (2.61)where ? = (?1, . . . , ?n) and ? = (?1, . . . , ?m). Substituting equations (2.61) intou = f (x), one obtainsu? ? ??(x?, f (x?)) + O(?2) = f (x? ? ??(x?, f (x?)) + O(?2))= f (x?) ? ?n?i=1? f (x?)?x?i?i(x?, f (x?)) + O(?2). (2.62)It follows thatu? = f (x?) +????????(x?, f (x?)) ?n?i=1? f (x?)?x?i?i(x?, f (x?))??????? ? + O(?2). (2.63)Replacing x? by x, u? by u in (2.63), one obtains the image surfaces u = g(x; ?).Keeping x invariant, the following one-parameter group of transformationsx?i = xi, i = 1, . . . , n,u? j = u +?????????(x, u) ?n?k=1ujk?k(x, u)???????? ? + O(?2) j = 1, . . . ,m,(2.64)maps u = f (x) into the same image surfaces u = g(x; ?). It turns out that theinfinitesimal generator X and its evolutionary form ?X determine the same actionon surfaces. Figure 2.1 illustrates the action of exp(?X) and exp(? ?X).In general, a generalized vector field X and its evolutionary form ?X are equiv-alent in symmetry analysis [21, 25, 29, 39, 53, 75].Theorem 2.2.42 A generalized vector field X is an infinitesimal symmetry of aDE system if and only if its evolutionary form ?X is.242.2. Symmetries of DE systems(a) The action of exp(?X) (b) The action of exp(? ?X)Figure 2.1: The action of exp(?X) and exp(? ?X).One can extend Lie?s algorithm for finding point symmetries to the algorithmfor finding local symmetries of a DE system through replacing the symmetry by itsevolutionary form and letting the characteristics depend on x, u and a fixed orderof derivatives. Then one applies its infinite prolongation to the given DE system.In eliminating the dependencies among the derivatives of u, it is necessary to takethe differential consequences of the given DE system into consideration.Symmetry is one of the main tools in the analysis of DEs. In the ODE case,using a continuous symmetry to integrate an ODE is one of the most importantapplications. It turns out that one can use a continuous symmetry to reduce theorder of a given ODE. If an ODE has a one-parameter symmetry group, then theorder of the ODE can be reduced by one. If an ODE has an r-parameter symmetrygroup and its corresponding Lie algebra is solvable, the order of the ODE canbe reduced by r. Moreover, one can obtain the solutions of a given ODE from thesolutions of the corresponding reduced ODE. The reduction of the order of an ODEthrough a symmetry can be obtained in two different ways: canonical coordinatesor differential invariants.In the PDE case, besides constructing new solutions from known ones, one canuse symmetries to construct invariant solutions, which will be discussed in Chap-ter 5. Moreover, one can use the knowledge of symmetries to construct invertiblemappings relating PDEs. In [62] (also see [25, 30, 31]), Kumei and Bluman intro-duced an algorithm to determine whether there exists a smooth invertible mappingthat maps a nonlinear PDE system to a linear PDE system based on symmetries252.2. Symmetries of DE systemsand find such a mapping when it exists. In [17, 18] (also see [25, 30]), Blumanpresented a symmetry-based algorithm to determine whether a linear PDE withvariable coefficients can be invertible mapped to a linear PDE with constant coef-ficients and find such a mapping when it exists.2.2.6 Equivalence transformations and symmetry classificationIf a given DE system involves some constitutive functions and/or parameters, thereexist some special transformations of the system, which preserve the differentialstructure of the DEs in the family. Such transformations are important in the sym-metry analysis of the system. Consider a family FK of DE systems R{x, u; K}:R?(x, u(l); K) = 0, ? = 1, . . . , s, (2.65)involving p constitutive functions and/or parameters K = (K1, . . . ,Kp).Definition 2.2.43 An equivalence transformation of a family FK of DE systemsis a transformation that maps a DE system R{x, u; K} ? FK to another DE system?R{x?, u?; ?K} ? FK .For example, a one-parameter group of equivalence transformations of a familyFK of DE systems is a one-parameter group of equivalence transformationsx?i = ?i(x, u; ?), i = 1, . . . , n,u? j = ? j(x, u; ?), j = 1, . . . ,m,?K? = ??(x, u,K; ?), ? = 1, . . . , p,(2.66)which maps a DE system R{x, u; K} ? FK to another DE system ?R{x?, u?; ?K} ? FK .Example 2.2.44 Consider the nonlinear reaction-diffusion equationut ? uxx = Q(u), (2.67)where Q(u) is an arbitrary constitutive function. In order to find the one-parametergroups of equivalence transformations of the nonlinear reaction-diffusion equation(2.67), one treats the constitutive function Q(u) as a new dependent variable andapply Algorithm 2.2.35 to the nonlinear reaction-diffusion equation (2.67). Ac-cording to the transformations (2.66), one can suppose the infinitesimal generatoris given byX = ?(x, t, u) ??x+ ?(x, t, u) ??t+ ?(x, t, u) ??u+ ?(x, t, u,Q) ??Q . (2.68)262.2. Symmetries of DE systemsApplying (2.68) to the nonlinear reaction-diffusion equation (2.67), one obtains theinfinitesimal generators given byX1 = uF??u+ (QF + uF?) ??Q ,X2 = G??u+ (Gt ? Gxx) ??Q ,X3 = 2H??x ? xuH? ??u ? x(uH?? + QH?) ??Q ,X4 =12xP???x + P??t ?18 x2uP????u+(14uP?? ? 18x2uP??? ? 18x2QP?? ? QP?)??Q ,(2.69)where F = F(t), G = G(x, t), H = H(t) and P = P(t) are arbitrary functions.Since the constitutive function Q depends only on u, for the above Xi to generateequivalence transformations, F, G, H and P must satisfy F(t) = C1, G(x, t) = C2,H? = 0 and P?? = 0, where C1 and C2 are constants. Therefore, the infinitesimalgenerators (2.69) becomeY1 = u??u + Q??Q ,Y2 =??u , Y3 =??x , Y4 =??tY5 = 2t??t+ x ??x? 2Q ??Q .(2.70)The five-parameter group of equivalence transformations arising from the infinites-imal generators (2.70) is given byx? = a1x + a2,?t = a21t + a3,u? = a4u + a5,?Q(u?) = a4a21Q(u).(2.71)A symmetry classification problem of a family FK of DE systems with consti-tutive functions and/or parameters is to classify DE systems in the family into somesubfamilies with the property that all DE systems in the same subfamily admit thesame symmetries. It is common to use the group of equivalence transformations tosimplify the point symmetry classification problem [1, 2, 56, 68, 80, 82]. There-fore, a point symmetry classification table is usually presented modulo the groupof equivalence transformations of the given family of DE systems.272.3. Conservation laws and the direct method2.3 Conservation laws and the direct method2.3.1 Conservation lawsConsider a DE system R{x; u} with n independent variables x = (x1, . . . , xn) and mdependent variables u = (u1, . . . , um).Definition 2.3.1 A conservation law of R{x; u} is a divergence expressiondiv (?[u]) = D1?1[u] + ? ? ? + Dn?n[u] = 0 (2.72)holding for every solution u = f (x) of R{x; u}. The functions ?i[u], i = 1, . . . , n,are called the fluxes of the CL.Remark 2.3.2 If one of the independent variables of a PDE system is time t, a CLof the PDE system is of the formDt?[u] +n?i=1Di?i[u] = 0, (2.73)where x = (x1, . . . , xn) are n spatial variables; ?[u] is the density of the CL (2.73),and ?i[u], i = 1, . . . , n, are the spatial fluxes of the CL (2.73).A CL could trivially hold in two different cases.(1) The n-tuple ?[u] in (2.72) vanishes for all solutions of the given DE system.This type of triviality is called the first kind of triviality.(2) The divergence expression div(?[u]) ? 0, i.e., div(?[u]) = 0 holds for allfunctions u = f (x). This type of triviality is called the second kind of trivi-ality. Such n-tuple ?[u] yields a null divergence.There is a useful characterization of a null divergence as seen in the followingtheorem.Theorem 2.3.3 Suppose ?[U] = (?1[U], . . . ,?n[U]) is an n-tuple of smoothfunctions depending on x, U = (U1, . . . ,Um) and their derivatives. Then ?[U]yields a null divergence if and only if there exist smooth functions Qi j[U], i, j =1, . . . , n, such thatQi j[U] = ?Q ji[U], i, j = 1, . . . , n, (2.74)and?i[U] =n?j=1D jQi j[U], i = 1, . . . , n, (2.75)hold for all functions U(x) = (U1(x), . . . ,Um(x)).282.3. Conservation laws and the direct methodProof. See [75] for the proof.Definition 2.3.4 A CL div (?[u]) = 0 of a given PDE system is trivial if its fluxesare of the form ?i[u] = Ai[u] + Bi[u], where Ai[u] are the fluxes of a first kindtrivial CL and Bi[u] are the fluxes of a second kind trivial CL, i = 1, . . . , n. TwoCLs are equivalent if they differ by a trivial CL.Throughout this thesis, unless stated otherwise, a ?CL? means a ?nontrivialCL?.A set of ? CLs{div(? j[u])= 0}?j=1 is said to be linearly dependent if thereexists a set of constants{a j}?j=1, not all zero, such that the linear combinationdiv???????????j=1a j? j[u]????????? = 0 is a trivial CL.Consider a totally nondegenerate DE system R{x; u} with n independent vari-ables x = (x1, . . . , xn) and m dependent variables u = (u1, . . . , um) given byR?[u] = R?(x, u(l)) = 0, ? = 1, . . . , s. (2.76)Definition 2.3.5 A multiplier (characteristic) of a CL div(?[u])= 0 of the DEsystem R{x; u} (2.76) is an s-tuple ?[U] = (?1[U], . . . ,?s[U]) such thats??=1??[U]R?[U] ? div(?[U]) (2.77)holds for all functions U(x). A multiplier is trivial if it vanishes for all solutions ofthe DE system. Two multipliers are equivalent if they differ by a trivial multiplier.In fact, any CL of a totally nondegenerate DE system arises from multipliersto within a trivial CL. Since the DE system R{x; u} (2.76) is totally nondegenerate,div(?[u]) = 0 is a CL of the system if and only if there exist functions QJ? suchthatdiv(?[U]) ???,JQJ?[U]DJR?[U] (2.78)holds for all functions U(x). After integrating by parts, one obtainsdiv(?[U]) ? Q[U] ? R[U] + div(?[U]), R = (R1, . . . ,Rs), (2.79)where Q[U] = (Q1[U], . . . ,Qs[U]) with entries Q?[U] =?J(?D)J QJ?[U], and?[U] = (?1[U], . . . ,?n[U]) depends linearly on R?[U] and their derivatives.Therefore, div(?[u]) = 0 is a trivial CL of the given system.292.3. Conservation laws and the direct method2.3.2 The direct methodDefinition 2.3.6 The Euler operator with respect to u?, 1 ? ? ? m, is defined byEu? =?J(?D)J ??u?J, (2.80)where the summation is over all multi-indices J = ( j1, . . . , jk) with 1 ? j? ? nfor ? = 1, . . . , k, |J | ? 0, and (?D)J = (?1)kDJ = (?D j1 )(?D j2 ) ? ? ? (?D jk ) forJ = ( j1, . . . , jk).One of the most important properties of the Euler operators is characterized inthe following theorem [9, 25, 75].Theorem 2.3.7 The equations EU?(F[U]) ? 0, ? = 1, . . . ,m, hold for arbitraryfunction U(x) if and only if F[U] = div(?[U]) for some m-tuple of differentialfunctions ?[U] = (?1[U], . . . ,?m[U]).In [9] (also see [25]), Anco and Bluman introduce a systematic way, called thedirect method, for constructing CLs for the DE system R{x; u} (2.76).Algorithm 2.3.8 (The direct method for constructing CLs):1. Let ??[U] = ??(x,U(k)), ? = 1, . . . , s, be the k-th order multipliers for CLsof the DE system R{x; u} (2.76). Eliminate the dependence of the derivativesof U due to the DE system R{x; u} (2.76).2. Solve the system of determining equations generated byEU?????????s??=1??[U]R?[U]???????? ? 0, ? = 1, . . . ,m, (2.81)explicitly to obtain its general solution sets {??[U]}s?=1.3. Find the corresponding fluxes for each solution set {??[U]}s?=1. In particu-lar, find an n-tuple of functions ?[U] = (?1[U], . . . ,?n[U]) satisfying theidentitydiv(?[U]) ?s??=1??[U]R?[U]. (2.82)4. Replace U and their derivatives by u and their derivatives in (2.82) to obtaina CL of the DE system R{x; u} (2.76) given bydiv(?[u]) = 0. (2.83)302.3. Conservation laws and the direct methodRemark 2.3.9 For a PDE system with n independent variables (t, x) = (t, x1, . . . ,xn?1) and m dependent variables (u1, . . . , um) in Cauchy-Kovalevskaya form:?l j u j?tl j= f j(x, t, u, ?u, . . . , ?lu), 1 ? l j ? l, j = 1, . . . ,m, (2.84)where, for each j, the orders of all derivatives with respect to t appearing in f j arelower than l j, it suffices to consider specific forms of multipliers. In particular, onecan rewrite the PDE system (2.84) into the following equivalent form?u???t =?f ?(t, x, u?, ?xu?, . . . , ?lxu?), ? = 1, . . . ,m?, (2.85)where ?ixu? denotes the i-th order partial derivatives of u? with respect to x (?xu? =?1xu?). It follows that all derivatives with respect to t can be expressed in terms of t,x, u? and derivatives of u? with respect to x from the equations (2.85). According toTheorem 2.3.10, all nontrivial local CLs of (2.85), up to equivalence classes, arisefrom the multipliers of the form {??[ ?U] = ??(x, t, ?U, ?x ?U, . . . , ?kx ?U)}m??=1.The correspondence between equivalence classes of CLs and those of multi-pliers of a PDE system in Cauchy-Kovalevskaya form is stated in the followingtheorem [9, 25, 75].Theorem 2.3.10 Suppose a given PDE system is in Cauchy-Kovalevskaya form.Let L and ?L be two CLs of the given PDE system determined from the multipliers? and ??, respectively. Then L and ?L are equivalent CLs if and only if ? and ?? areequivalent multipliers.Once one obtains the set of multipliers for a local CL, a problem is how to findthe corresponding fluxes. The method that is used to find fluxes depends on specificproblems [4, 6, 8, 9, 42, 93]. For multipliers in simple forms, an effective wayis integration by parts. For multipliers in complicated forms, Anco and Blumanintroduced the following integral formula for finding the corresponding fluxes [9].Theorem 2.3.11 For a given set of local CL multipliers {??[U] = ??(x,U(k))}s?=1of the DE system (2.76), its corresponding fluxes are given by the following inte-gral formulas:?i[U] = ?i[ ?U]+? 10(S i[U ? ?U,?[?U + (1 ? ?) ?U]; R[?U + (1 ? ?) ?U]]+ ?S i[U ? ?U,R[?U + (1 ? ?) ?U];?[?U + (1 ? ?) ?U]])d?,i = 1, . . . , n,(2.86)312.3. Conservation laws and the direct methodwithS i[V,W; R[U]] =l?1?p=0l?p?1?q=0m??=1?I,J(?1)qDIV?DJ????????s??=1W??R?[U]?U?J,i,I???????? , (2.87)and?S i[ ?V , ?W;?[U]] =k?1?p=0k?p?1?q=0m??=1?I,J(?1)qDI ?V?DJ????????s??=1?W????[U]?U?J,i,I???????? , (2.88)for arbitrary functions V = (V1(x), . . . ,Vm(x)), W = (W1(x), . . . ,Ws(x)), ?V =( ?V1(x), . . . , ?Vm(x)) and ?W = ( ?W1(x), . . . , ?Ws(x)), where J = ( j1, . . . , jq) and I =(i1, . . . , ip) are ordered multi-indices such that 1 ? j1 ? ? ? ? ? jq ? i ? i1 ? ? ? ? ?ip ? n.Example 2.3.12 Consider the nonlinear diffusion equationut ? (uux)x = 0. (2.89)Applying the Euler operator EU to the function ?(x, t,U) (Ut ? (UUx)x) yields theexpressionEU (?(x, t,U) (Ut ? (UUx)x)) ? 0, (2.90)for arbitrary U(x, t). Splitting equation (2.90) with respect to arbitrary Ut, Ux andUxx leads to? ?U ? U?UU = 0,? 2U?U = 0,? 2U?xU = 0,? ?t ? U?xx = 0.(2.91)The general solution of equations (2.91) is ?(x, t,U) = c1x + c2, where c1 and c2are arbitrary constants. Thus there are two linearly independent multipliers of theform ? = ?(x, t,U): ?1 = 1 and ?2 = x. The corresponding CLs are given by?1 = 1 : Dtu ? Dx(uux) = 0,?2 = x : Dt(xu) ? Dx(xuux ?u22)= 0.(2.92)The next theorem shows that it is possible to use the direct method to find alllocal CLs for an evolution equation of specific form [54].322.3. Conservation laws and the direct methodTheorem 2.3.13 Consider the (1+1)-dimensional scalar evolution equation withtwo independent variables (x, t) and one dependent variable u of even order 2lgiven byut = F(x, t, u, ?xu, . . . , ?2lx u). (2.93)If a CL of the equation (2.93) is given byDt?[u] + Dx?[u] = 0, (2.94)then the maximal order of a derivative of u in ?[u] is l.A notable result relating CLs and symmetry groups was obtained by Noether[73]. Noether showed that each CL of a DE system admitting a variational princi-ple arises from a point symmetry (variational symmetry) of the action functional.Boyer [38] extended Noether?s theorem by taking higher-order symmetries intoconsideration. In particular, a one-parameter higher-order transformation (in evolu-tionary form) is a variational symmetry of an action functional if the correspondingLagrangian is invariant to within a divergence term under such a transformation.However, there are some limitations of Noether?s theorem for finding CLs of agiven DE system. First of all, the given DE system is restricted to Euler-Lagrangeequations for some variational problem. In addition, it is sometimes difficult todetermine the variational symmetry for a given system of Euler-Lagrange equa-tions. It is incorrect that all symmetries of a system of Euler-Lagrange equationsare variational symmetries of the corresponding action functional. In order tocheck whether a symmetry of a system of Euler-Lagrange equations is a varia-tional symmetry of the corresponding action functional, one must firstly determinethe corresponding Lagrangian of the action functional. Finally, Noether?s theoremis coordinate-dependent since an invertible transformation may transform a DEsystem admitting a variational principle to a DE system that has no such property.However, any invertible transformation maps a CL of a given DE system to a CL ofthe transformed one, i.e., CLs are coordinate-independent. Thus an ideal methodfor finding CLs should be coordinate-independent.The direct method for finding CLs is superior to Noether?s theorem in the sensethat it is free of all the above limitations. The direct method can be applied to anyDE system, whether it is variational or not. Moreover, it does not require one tofind the variational symmetries and Lagrangian when the given DE system is a sys-tem of Euler-Lagrange equations. In fact, the direct method directly generates themultipliers for CLs of any given DE system. Most importantly, the direct methodis coordinate-independent.The next theorem shows that a divergence expression is mapped to a divergenceexpression by a point transformation [25, 35, 81].332.3. Conservation laws and the direct methodTheorem 2.3.14 Under the point transformationxi = xi(z,W), i = 1, . . . , n,U j = U j(z,W), j = 1, . . . ,m, (2.95)where U = (U1(x), . . . ,Um(x)), z = (z1, . . . , zn) and W = (W1(z), . . . ,Wm(z)), thereexists an n-tuple ?[W] = (?1[W], . . . ,?n[W]) such thatJ[W]div(?[U]) = d?iv(?[W]), (2.96)where ?[U] = (?1[U], . . . ,?n[U]), J[W] = D(x1 ,...,xn)D(z1,...,zn) and d?iv is the divergenceoperator on (z,W) space.Remark 2.3.15 Similar to Theorem 2.3.14, it is straightforward to show that ifm = 1, then any contact transformation maps a divergence expression into a diver-gence expression.The following theorem illustrates how the fluxes change under a symmetry inevolutionary form [75].Theorem 2.3.16 Let div(?[u]) = 0 be a CL of a totally nondegenerate DE sys-tem R{x; u}. If ?X is a symmetry in evolutionary form of R{x; u}, then the in-duced n-tuple ?? = ?X(?)(?[u]), with entries ??i[u] = ?X(?)(?i[u]), also yields aCL: div( ??[u]) = 0.Besides the basic applications of CLs, a CL could also be used to determinewhether a nonlinear PDE can be mapped into a linear PDE through an invertibletransformation. In [10], Anco, Bluman and Wolf presented an algorithm for deter-mining whether there exist an invertible transformation that maps a nonlinear PDEinto a linear PDE through CLs. Moreover, one can use this algorithm to explicitlyfind such an invertible transformation provided it exists.34Chapter 3Nonlocally Related PDE Systemsand Applications3.1 IntroductionIn Chapter 2, we presented algorithms to find local (point, contact or higher-order)symmetries for a given DE system. Since a symmetry of a DE system is a trans-formation that keeps its solution manifold invariant, it is possible that the infinites-imals of a continuous symmetry need not depend only on independent and depen-dent variables and their derivatives. Such symmetries are not local symmetries,and are called nonlocal symmetries. A special kind of nonlocal symmetry is onewhose infinitesimals depend on integrals of the dependent variables. However, it isnot possible to find such nonlocal symmetries through a direct application of Lie?salgorithm to the given DE system. In addition, there is the problem of how to usesuch symmetries.As stated in Chapter 1, two PDE systems are equivalent and nonlocally relatedif they have the following properties.(1) Any solution of either PDE system yields a solution of the other PDE system.(2) The solutions of either PDE system yield all solutions of the other PDE sys-tem.(3) The correspondence between the solutions of these two PDE systems is notone-to-one.Nonlocally related PDE systems are important in the analysis of a given PDEsystem. In particular, one may be able to obtain new exact solutions for a givenPDE system through solutions of its nonlocally related PDE systems. Bluman,Kumei and Reid [32] introduced a systematic CL-based method for constructingnonlocally related PDE systems of a PDE system with two independent variables.In [23], an extended procedure for the construction of a tree of nonlocally relatedPDE systems was presented. It turns out that one can obtain nonlocal symme-tries and nonlocal CLs for a given PDE system through its nonlocally related PDEsystems (see [25] and references therein).353.1. IntroductionIn [7], a systematic CL-based method for constructing nonlocally related PDEsystems of a PDE system with three or more independent variables was presented.However, in this case, nonlocally related PDE systems arising from divergence-type CLs are invariant under gauge transformations, hence are under-determined.But it turns out that one can also obtain nonlocal CLs from such nonlocally re-lated PDE systems. Unlike the situation for two independent variables, it is shownthat, in the case of three or more independent variables, nonlocally related PDEsystems arising from divergence-type CLs cannot yield nonlocal symmetries of agiven PDE system. In order to find nonlocal symmetries of a given PDE system,it is necessary to add gauge constraints to such nonlocally related PDE systems.Conservation laws of a PDE system with three or more independent variables arenot limited to divergence-type CLs. There exist curl-type CLs (lower-degree CLs)of a PDE system with three or more independent variables. A systematic methodfor constructing nonlocally related PDE systems of a PDE system with three ormore independent variables through lower-degree CLs is presented in [25, 43, 44].In addition, it is shown that CLs with degree one can yield determined potentialsystems.In this chapter, we present the known CL-based method for constructing non-locally related PDE systems (potential systems) and the method for constructingsubsystems. We also state the method for seeking nonlocal symmetries and nonlo-cal CLs of a given PDE system through its nonlocally related PDE systems. More-over, at the end of this chapter, we investigate relationships between symmetries ofa given PDE system and those of its potential systems.Two new results are presented in this chapter. Consider a given PDE sys-tem R{x, t; u} with two independent variables (x, t) and m dependent variablesu = (u1, . . . , um).? It is shown that for two potential systems S1{x, t; u, v} and S2{x, t; u,w} ofR{x, t; u} written in Cauchy-Kovalevskaya form, arising from two nontriviallinearly independent local CLs of R{x, t; u}, the potential variable of onesystem cannot be expressed as a local function in terms of the independentvariables, dependent variables and their derivatives of the other system.? It is shown that if R{x, t; u} has precisely a finite number n of local CLs, thenany local symmetry of R{x, t; u} can be obtained by projection from a localsymmetry of its corresponding n-plet potential system.363.2. CL-based method for constructing nonlocally related PDE systems in 2D3.2 CL-based method for constructing nonlocally relatedPDE systems in 2D3.2.1 Potential systems and subsystemsConsider a PDE system R{x, t; u} with two independent variables (x, t) and m de-pendent variables u = (u1, . . . , um) given byR?[u] = 0, ? = 1, . . . , s. (3.1)In [32], Bluman, Kumei and Reid introduced a systematic way to constructnonlocally related PDE systems of a PDE system R{x, t; u} (3.1) based on its CLs.Suppose R{x, t; u} (3.1) has a nontrivial CL given byDt?[u] + Dx?[u] = 0. (3.2)By introducing a potential variable v, one obtains a pair of potential equationsvx = ?[u],vt = ??[u].(3.3)Definition 3.2.1 A system of PDEs S{x, t; u, v} consisting of the given PDE systemR{x, t; u} and the pair of potential equations (3.3) arising from a CL of R{x, t; u} isa potential system of R{x, t; u}.Remark 3.2.2 If the given PDE system R{x, t; u} is a scalar PDE and the CL (3.2)arises from multipliers that do not involve u and its derivatives, it is redundantto add R{x, t; u} to the potential system S{x, t; u, v}. One can deduce the PDE inR{x, t; u} from the pair of potential equations through integrability conditions.Remark 3.2.3 The given PDE system R{x, t; u} and its potential system S{x, t; u, v}are equivalent. Without loss of generality, one can consider the case when thegiven PDE system R{x, t; u} is a scalar PDE. Suppose u = f (x, t) is a solution ofR{x, t; u}. Since Dt?[u] + Dx?[u] = 0 is a CL of R{x, t; u}, due to the integrabilitycondition vxt = vtx, there exists a function g(x, t) such that (u, v) = ( f (x, t), g(x, t))is a solution of S{x, t; u, v}. Thus any solution of R{x, t; u} yields a solution ofS{x, t; u, v}. Conversely, if (u, v) = ( f (x, t), g(x, t)) is a solution of S{x, t; u, v},by projection, u = f (x, t) solves R{x, t; u}. Hence R{x, t; u} and S{x, t; u, v} areequivalent. Moreover, if (u, v) = ( f (x, t), g(x, t)) is a solution of S{x, t; u, v}, so is(u, v) = ( f (x, t), g(x, t) + C), where C is arbitrary constant. It follows that the re-lationship between the solutions of the given PDE system R{x, t; u} and those ofits potential system S{x, t; u, v} is not one-to-one. Hence the given PDE systemR{x, t; u} and its potential system S{x, t; u, v} are nonlocally related.373.2. CL-based method for constructing nonlocally related PDE systems in 2DRemark 3.2.4 The potential variable v is a nonlocal variable of R{x, t; u}, i.e., vcannot be expressed as a local function of the variables in R{x, t; u} and their deriva-tives. Suppose v is a local variable of R{x, t; u}, then v = F[u] on the solutions ofR{x, t; u} for some local function F. Sincevx = ?[u],vt = ??[u],it follows thatDx(F[u]) = ?[u],Dt(F[u]) = ??[u],on the solutions of R{x, t; u}. Consequently, on the solutions of R{x, t; u}, the CL(3.2) can be rewritten intoDt?[u] + Dx?[u] = Dt (Dx(F[u])) + Dx (Dt(F[u])) = 0,which implies that the CL (3.2) is a trivial CL. This contradicts the assumption thatthe CL (3.2) is nontrivial. Hence v is a nonlocal variable.Since each potential system arising from a nontrivial CL is nonlocally relatedto the given PDE system, we also use the terminology nonlocally related CL-basedsystem (nonlocally related CL system) to denote a potential system.Example 3.2.5 Consider the nonlinear diffusion equationut = (K(u)ux)x . (3.4)The nonlinear diffusion equation (3.4) is in a CL form. By introducing a potentialvariable, one obtains the potential system given byvx = u,vt = K(u)ux.(3.5)Moreover, the nonlinear diffusion equation (3.4) has another CL given by(xu)t ? (x(L(u))x ? L(u))x = 0, (3.6)where L?(u) = K(u). Based on the CL (3.6), one can construct another potentialsystem of the nonlinear diffusion equation (3.4) given by?x = xu,?t = x(L(u))x ? L(u).(3.7)383.2. CL-based method for constructing nonlocally related PDE systems in 2DFor potential systems arising from equivalent CLs, the following theorem showsthat such potential systems are locally related [25].Theorem 3.2.6 If two potential systems S1{x, t; u, v} and S2{x, t; u,w} of a givenPDE system R{x, t; u} arise from two equivalent CLs of R{x, t; u}, then S1{x, t; u, v}and S2{x, t; u,w} are locally related. In particular, w = v + F[u] for some functionF[u].The following new theorem concerns the relationship between two potentialvariables arising from two nontrivial and linearly independent local CLs.Theorem 3.2.7 Suppose two potential systems S1{x, t; u, v} and S2{x, t; u,w} of agiven PDE system R{x, t; u} arise from two nontrivial and linearly independentlocal CLs, where v and w are potential variables. If S1{x, t; u, v} and S2{x, t; u,w}are in Cauchy-Kovalevskaya form, then w is a nonlocal variable of S1{x, t; u, v} andv is a nonlocal variable of S2{x, t; u,w}.Proof. In order to show w is a nonlocal variable of S1{x, t; u, v}, it suffices to showthat w cannot be expressed as a local function of the variables in S1{x, t; u, v} andtheir derivatives. Without loss of generality, one can assume that R{x, t; u} is ascalar PDE: R[u] = 0. Suppose S1{x, t; u, v} arises from the CLDt?1[u] ? Dx?1[u] = 0, (3.8)with corresponding multiplier ?1 = ?1[U], i.e.,?1[U]R[U] = Dt?1[U] ? Dx?1[U], (3.9)for arbitrary U. Since S1{x, t; u, v} is in Cauchy-Kovalevskaya form, S1{x, t; u, v} isgiven byvx = ?1[u],vt = ?1[u].(3.10)Suppose S2{x, t; u,w} arises from the CLDt?2[u] ? Dx?2[u] = 0, (3.11)with corresponding multiplier ?2, i.e.,?2[U]R[U] = Dt?2[U] ? Dx?2[U], (3.12)for arbitrary U. Since S2{x, t; u, v} is in Cauchy-Kovalevskaya form, S2{x, t; u,w}is given bywx = ?2[u],wt = ?2[u].(3.13)393.2. CL-based method for constructing nonlocally related PDE systems in 2DAssume w can be expressed by a local function of the variables in S1{x, t; u, v} andtheir derivatives, i.e., w = F[u, v] for some local function F. Then the CL (3.11) isa trivial CL of S1{x, t; u, v}, sinceDt?2[u] ? Dx?2[u] = Dt(DxF[u, v]) ? Dx(DtF[u, v]) (3.14)on solutions of S1{x, t; u, v}.On the other hand,Dt?2[U] ? Dx?2[U] = ?2R[U] =?2?1(Dt(?1[U] ? Vx) ? Dx(?1[U] ? Vt))= Dt(?2?1(?1[U] ? Vx))? Dx(?2?1(?1[U] ? Vt))? Dt(?2?1)(?1[U] ? Vx) + Dx(?2?1)(?1[U] ? Vt),(3.15)for arbitrary U and V . The identity (3.15) implies the multipliers for the CL (3.11)with respect to S1{x, t; u, v} are Dt(?2?1)and ?Dx(?2?1).Theorem 2.3.10 shows that for a PDE system in Cauchy-Kovalevskaya form, aCL is trivial if and only if its multipliers are trivial. Since the CL (3.11) is a secondkind trivial CL of S1{x, t; u, v}, it follows that Dt(?2?1)? 0 and ?Dx(?2?1)? 0.Consequently, ?2 = c?1 for some constant c. Hence the CLs (3.8) and (3.11) arelinearly dependent. It turns out that w is not a local function of the variables inS1{x, t; u, v} and their derivatives. Thus w is a nonlocal variable of S1{x, t; u, v}.Similarly, one can show that v is a nonlocal variable of S2{x, t; u,w}. If the PDE system R{x, t; u} (3.1) has n linearly independent local CLs:Dt?i[u] + Dx?i[u] = 0, i = 1, . . . , n, (3.16)one can introduce n potential variables vi with the potential equations:vix = ?i[u],vit = ??i[u].(3.17)Let Pi denote the potential equations (3.17). Through the potential equations(3.17), one can obtain n potential systems S(1){x, t; u, vi} = R{x, t; u} ? Pi.Definition 3.2.8 A k-plet potential system (1 ? k ? n) of a PDE system R{x, t; u}with n linearly independent local CLs is the potential systemS(k){x, t; u, vi1 , . . . , vik } = R{x, t; u} ? Pi1 ? ? ? ? ? Pik . (3.18)In particular, for k = 1, 2, 3, 4, such k-plet potential systems are called singlets,couplets, triplets and quadruplets, respectively.403.2. CL-based method for constructing nonlocally related PDE systems in 2DExample 3.2.9 Consider the nonlinear diffusion equation (3.4). A couplet poten-tial system of (3.4) is given byvx = u,vt = K(u)ux,?x = xu,?t = x(L(u))x ? L(u).(3.19)For a PDE system R{x, t; u} (3.1) with n linearly independent local CLs, onecan construct 2n ? 1 potential systems:? n singlets: S(1){x, t; u, vi}, i = 1, . . . , n.? 12n(n ? 1) couplets: S(2){x, t; u, vi, v j} i, j = 1, . . . , n and i , j.?...? One n-plet: S(n){x, t; u, v1, . . . , vn}.Definition 3.2.10 For a PDE system R{x, t; u} with n linearly independent localCLs, the set of all 2n ? 1 potential systems arising from n potential variables v1,. . . , vn is called a combination potential system, denoted by Pv1...vn .3.2.2 SubsystemsAnother effective way to construct equivalent PDE systems of a PDE system R{x, t;u} with two independent variables (x, t) and m ? 2 dependent variables u = (u1, . . . ,um) is through excluding some dependent variables of R{x, t; u}.Definition 3.2.11 Consider a PDE system R{x, t; u} with two independent vari-ables (x, t) and m ? 2 dependent variables u = (u1, . . . , um). A subsystem ofR{x, t; u} is a PDE system obtained by excluding some dependent variables ofR{x, t; u} and has the properties:(1) Any solution of the subsystem yields a solution of R{x, t; u}.(2) The solutions of the subsystem yield all solutions of R{x, t; u}.Example 3.2.12 Consider the potential system (3.5) of the nonlinear diffusionequation (3.4). By excluding u from (3.5), one obtains the subsystem of (3.5)given byvt = K(vx)vxx. (3.20)413.2. CL-based method for constructing nonlocally related PDE systems in 2DExample 3.2.13 Consider the Lagrange system of gas dynamics given byqx ? vy = 0,vx + py = 0,px + B(p, q)vy = 0,(3.21)where B(p, q) is the constitutive function. By excluding v from (3.21), one obtainsthe subsystem given byqxx + pyy = 0,px + B(p, q)qx = 0.(3.22)The following theorem states when a subsystem of R{x, t; u} is nonlocally re-lated to R{x, t; u}.Theorem 3.2.14 A subsystem ?R{x, t; u1, . . . , um?1}, obtained by excluding the de-pendent variable um from the PDE system R{x, t; u}, is nonlocally related to R{x, t;u} if and only if um cannot be directly expressed from the PDEs of R{x, t; u} interms of x, t, the dependent variables u1, . . . , um?1 of ?R{x, t; u1, . . . , um?1} andtheir derivatives. Otherwise the subsystem ?R{x, t; u1, . . . , um?1} is locally related toR{x, t; u}.Proof. See [23, 25] for the proof.From Theorem 3.2.14, one concludes that the PDE (3.20) is a locally relatedsubsystem of the PDE system (3.5), since the excluded variable u can be expressedin terms of x, t, v and its derivatives from the PDEs of the PDE system (3.5). Inparticular, u = vx. But the PDE system (3.22) is a nonlocally related subsystem ofthe Lagrange system of gas dynamics (3.21), since the excluded variable v cannotbe expressed as a local function of x, y, p, q and their derivatives from the PDEs ofthe PDE system (3.21).3.2.3 Procedure for constructing a tree of nonlocally related PDEsystemsFor a given PDE system R{x, t; u}, a basic procedure for the construction of a treeof nonlocally related PDE systems is as follows.Procedure 3.2.15 (A Tree Construction Procedure)1. Construction of potential systems. For each CL of R{x, t; u}, one intro-duces a potential variable. If n linearly independent CLs are found, one canconstruct 2n?1 potential systems. This yields up to 2n?1 nonlocally relatedPDE systems. Denote the resulting tree by T1.423.2. CL-based method for constructing nonlocally related PDE systems in 2D2. Continuation of construction of potential systems. For each potential sys-tem in T1, find its CLs using any method. Repeat Step 1 for such potentialsystems to obtain potential systems of each potential system. Repeat thisstep to obtain more potential systems. This leads to a tree T2 containingnonlocally related PDE systems.3. Construction of subsystems. For each PDE system in T3, by excluding itsdependent variables one by one when possible to generate its subsystems.Eliminate locally related PDE systems. This step could result in a larger treeof nonlocally related PDE systems denoted by T3.Remark 3.2.16 It is redundant to construct potential systems of the new resultingsubsystems in T3 for the reason that the set of all local CLs of a PDE systemincludes all local CLs of its subsystems [25].Remark 3.2.17 It may be difficult to determine whether two such systems are non-locally related. However, by construction, all PDE systems constructed by Proce-dure 3.2.15 are equivalent in the sense that the solutions of any such PDE systemcan be obtained from the solutions of any other such PDE system. Thus redundant(locally related) systems in a tree do not lead to incorrect results.Remark 3.2.18 Suppose the given PDE system R{x, t; u} has n (n ? 2) linearlyindependent CLs. Let vi, i = 1, . . . , n be the corresponding potential variables. It isshown that the linear combinations of such potential variables:w =ik?i=i1aivi, 1 ? i1 < i2 < ? ? ? < ik ? n, 1 ? k ? n, (3.23)could also yield potential systems that are nonlocally related to R{x, t; u} as well asS(1){x, t; u, vi} for arbitrary constants ai with at least two of them not zero [25, 57].It follows that a set of n CLs could yield a spectrum of singlet potential systems.Remark 3.2.19 In Procedure 3.2.15, subsystems are obtained by directly exclud-ing dependent variables of a given PDE system R{x, t; u}. In addition, it turns outone can employ hodograph transformations on a PDE system before excludingits dependent variables [20, 24, 25]. By excluding dependent variables from thetransformed PDE system, it is possible to generate additional nonlocally relatedsubsystems. More generally, any point transformation could be applied to a PDEsystem before excluding its dependent variables as long as one is able to excludesome dependent variables from the transformed PDE system.433.2. CL-based method for constructing nonlocally related PDE systems in 2DAccording to Remarks 3.2.18 and 3.2.19, it is straightforward to modify Steps2 and 3 in Procedure 3.2.15 to obtain further nonlocally related PDE systems in atree.For Step 2 in Procedure 3.2.15, there is an important result to avoid redundantcomputation in finding new CLs of a potential system [25, 26, 63].Theorem 3.2.20 A CL of any potential system S{x, t; u, v} is equivalent to a localCL of the given PDE system R{x, t; u} if and only if this CL arises from multipliersthat do not essentially depend on the potential variable v, modulo the equivalenceclass.According to Theorem 3.2.20, additional CLs of a potential system can onlyarise from multipliers that include the potential variable v. Therefore, it is neces-sary to consider multipliers with an essential dependence on at least one potentialvariable intoduced in Step 2 of Procedure 3.2.15.Example 3.2.21 Consider the nonlinear wave equationutt =(c2(u)ux)x, (3.24)where c(u) is an arbitrary constitutive function. In this example, we use Procedure3.2.15 to construct a tree of nonlocally related PDE systems of the nonlinear waveequation (3.24).Using the direct method, one can show that there are four multipliers of theform ? = ?(x, t,U) for arbitrary c(u) [24]. The corresponding CLs are given by?1 = 1 : utt ?(c2(u)ux)x= 0, (3.25)?2 = t : Dt (tut ? u) ? Dx(tc2(u)ux)= 0, (3.26)?3 = x : Dt (xut) ? Dx(xc2(u)ux ??c2(u)du)= 0, (3.27)and?4 = xt : Dt (x (tut ? u)) ? Dx(t(xc2(u)ux ??c2(u)du))= 0. (3.28)By introducing potential variables, one obtains four corresponding singlet potentialsystems given by ???????vx = ut,vt = c2(u)ux.(3.29)443.2. CL-based method for constructing nonlocally related PDE systems in 2D???????wx = tut ? u,wt = tc2(u)ux.(3.30)????????????x = xut,?t = xc2(u)ux ??c2(u)du. (3.31)and ????????????x = x (tut ? u) ,?t = t(xc2(u)ux ??c2(u)du).(3.32)The hodograph transformation???????x = x(u, v),t = t(u, v), (3.33)maps the potential system (3.29) into an invertibly equivalent linear PDE system???????xv = tu,xu = c2(u)tv.(3.34)In order to obtain more nonlocally related PDE systems, one seeks local CLs ofthe PDE system (3.34). If one considers the multipliers of the form (?1,?2) =(?1(u, v, X, T ),?2(u, v, X, T )), through the direct method, one can show that thereare only four multipliers holding for all c(u). These multipliers and their corre-sponding CLs are given by(?11,?21) = (1, 0) : xv ? tu = 0, (3.35)(?12,?22) = (0, 1) : xu ?(c2(u)t)v= 0, (3.36)(?13,?23) = (?X, T ) : (tx)u ?(x2 + c2(u)t22)v= 0, (3.37)and(?14,?24) = (?v, u) : (ux + vt)u ?(vx + uc2(u)t)v= 0. (3.38)From the above four CLs, one obtains four potential systems of the potential system(3.34) given by ???????????????????xv = tu,xu = c2(u)tv,pv = t,pu = x.(3.39)453.2. CL-based method for constructing nonlocally related PDE systems in 2D???????????????????xv = tu,xu = c2(u)tv,qv = x,qu = c2(u)t.(3.40)???????????????????????xv = tu,xu = c2(u)tv,rv = tx,ru =x2 + c2(u)t22.(3.41)and ???????????????????xv = tu,xu = c2(u)tv,?v = ux + vt,?u = vx + uc2(u)t.(3.42)Now we construct subsystems. After excluding the dependent variable x or tfrom the potential system (3.34), one obtains two subsystems:xvv =(c?2(u)xu)u, (3.43)andtuu = c2(u)tvv. (3.44)In [24], it was shown that the nonlinear wave equation (3.24), the potentialsystems (3.29)?(3.32), the PDEs (3.43) and (3.44) are mutually nonlocally related.Let T1 denote these nonlocally related PDE systems.One can show that the potential systems (3.39)?(3.42) are mutually nonlocallyrelated and nonlocally related to each PDE system in T1.Moreover, excluding the dependent variable x or t from the four potential sys-tems of the potential system (3.34) leads to additional nonlocally related PDE sys-tems. Take the PDE system (3.39) for example. Since the CL (3.35) is equivalentto the following CLDu(utu ? t) ? Dv(uc2(u)tv)= 0, (3.45)the PDE system (3.39) is locally related to the following PDE system???????????????????xv = tu,xu = c2(u)tv,?v = utu ? t,?u = uc2(u)tv.(3.46)463.3. Nonlocal symmetriesExcluding x from the PDE system (3.46), one obtains the following PDE system???????????????tuu =(c2(u)tv)v,?v = utu ? t,?u = uc2(u)tv.(3.47)In [24], it was shown that the PDE system (3.47) is nonlocally related to eachPDE system in T1. Moreover, one can show that the PDE system (3.47) is nonlo-cally related to potential systems (3.39)?(3.42).In summary, a tree of nonlocally related PDE systems involving the nonlinearwave equation (3.24) is shown in Figure 3.1.(3.29) ?? (3.34)(3.30)(3.31)(3.32)(3.24)(3.39) (3.40) (3.41) (3.42)(3.47)(3.43) (3.44)Figure 3.1: A tree of nonlocally related PDE systems for the nonlinear wave equa-tion (3.24).One can obtain a larger tree of nonloncally related PDE systems if one takesinto account the k-plet potential systems.3.3 Nonlocal symmetriesIn the previous section, we showed that one can obtain nonlocal CLs of a givenPDE system R{x, t; u} (3.1) from its potential systems. For example, the local CL(3.37) of the PDE system (3.34) is a nonlocal CL of the PDE (3.44). In particular,nonlocal CLs arising from potential systems must have multipliers involving atleast one potential variable. However, all local CLs of a subsystem of R{x, t; u}are local CLs of R{x, t; u} [25]. Analogous to nonlocal CLs, symmetries of a givenPDE system are not limited to local symmetries. Symmetries that are not local473.3. Nonlocal symmetriessymmetries are called nonlocal symmetries. How to find nonlocal symmetries is asignificant problem in symmetry analysis. Lie?s algorithm provides a simple andapplicable way to find local symmetries. However, there does not exist a uniformway to find nonlocal symmetries. In this section, we present a systematic procedureto seek nonlocal symmetries of R{x, t; u} from its nonlocally related PDE systems.It is shown that unlike nonlocal CLs, both potential systems and subsystems in atree of nonlocally related PDE systems can yield nonlocal symmetries of R{x, t; u}.Let S{x, t; u, v} with v = (v1, . . . , vk) be a k-plet potential system of the PDEsystem R{x, t; u} (3.1). Suppose S{x, t; u, v} has a point symmetry given by?????????????????????x? = x + ??(x, t, u, v) + O(?2),?t = t + ??(x, t, u, v) + O(?2),u?i = ui + ??i(x, t, u, v) + O(?2), i = 1, . . . ,m,v? j = v j + ?? j(x, t, u, v) + O(?2), j = 1, . . . , k,(3.48)with infinitesimal generatorX = ?(x, t, u, v) ??x+?(x, t, u, v) ??t+m?i=1?i(x, t, u, v) ??ui+k?j=1? j(x, t, u, v) ??v j. (3.49)The symmetry (3.48) leaves the solution manifold of S{x, t; u, v} invariant. Sincethe solution sets of S{x, t; u, v} and R{x, t; u} are equivalent, by projection, the one-parameter group of transformations (3.48) leads to a mapping that maps any solu-tion of R{x, t; u} to a solution of R{x, t; u}. Thus the one-parameter group of trans-formations (3.48) induces a symmetry of R{x, t; u} with corresponding infinitesimalgenerator?X = ?(x, t, u, v) ??x + ?(x, t, u, v)??t +m?i=1?i(x, t, u, v) ??ui. (3.50)If the infinitesimals (?(x, t, u, v), ?(x, t, u, v), ?i(x, t, u, v)) do not depend explic-itly on the nonlocal variable v, i.e., (?(x, t, u, v), ?(x, t, u, v), ?i(x, t, u, v)) = (?(x, t, u),?(x, t, u), ?i(x, t, u)), then ?X becomes?X = ?(x, t, u) ??x + ?(x, t, u)??t +m?i=1?i(x, t, u) ??ui, (3.51)which implies X only yields a point symmetry of R{x, t; u}.If the infinitesimals (?(x, t, u, v), ?(x, t, u, v), ?i(x, t, u, v)) essentially depend onthe nonlocal variable v, then the one-parameter group of transformations (3.48)defines a nonlocal symmetry of R{x, t; u}.483.3. Nonlocal symmetriesDefinition 3.3.1 The infinitesimal generator (3.50), obtained by projection of someinfinitesimal point symmetry of a k-plet potential system S{x, t; u, v} of R{x, t; u},generates a potential symmetry of R{x, t; u} if the infinitesimals of (3.50) dependexplicitly on one or more components of v.From the above discussion, one has proved the following theorem [25, 29, 32].Theorem 3.3.2 A potential symmetry of R{x, t; u} is a nonlocal symmetry of R{x,t; u}.Since the solution sets of all systems in a tree of related PDE systems are equiv-alent, local symmetries of any system yield symmetries of the other systems in thetree. However, for locally related subsystem (in the sense of Theorem 3.2.14), onehas the following theorem [25].Theorem 3.3.3 Any local symmetry of a locally related subsystem ?R{x, t; u1, . . . ,um?1} of a PDE system R{x, t; u} is a projection of some local symmetry of R{x, t; u}onto the variable space of ?R{x, t; u1, . . . , um?1}.The correspondence between the solutions of a given PDE system and thoseof its nonlocally related subsystems is not one-to-one. It is possible that nonlocalsymmetries of a given PDE system can arise from local symmetries of its nonlo-cally related subsystems.Example 3.3.4 Consider the nonlinear diffusion equation (3.4) for example. Thepoint symmetry classification of the nonlinear diffusion equation (3.4), modulo theequivalence transformations?t = a4t + a1,x? = a5x + a2,u? = a6u + a3,?K =a25a4K,(3.52)where are a1, . . . , a6 are arbitrary constants with a4a5a6 , 0, is presented in Table3.1 [79].The point symmetry classification of the potential system (3.5), modulo its493.3. Nonlocal symmetriesTable 3.1: Point symmetry classification for the nonlinear diffusion equation (3.4)K(u) # admitted point symmetriesarbitrary 3 X1 = ??x , X2 =??t , X3 = x??x + 2t??tu? (? , 0) 4 X1, X2, X3, X4 = x ??x + 2?u ??ueu 4 X1, X2, X3, X5 = x ??x + 2??uu?43 5 X1, X2, X3, X4 (? = ? 43 ), X6 = x2 ??x ? 3xu ??uequivalence transformations?t = a1t + a2,x? = a3x + a4v + a5,u? = a6 + a7ua3 + a4u,v? = a6x + a7v + a8,?K = (a3 + a4u)2a1K,(3.53)where a1, . . . , a8 are arbitrary constants with a1(a3a7 ? a4a6) , 0, is listed in Table3.2 [25, 82].Table 3.2: Point symmetry classification for the potential system (3.5)K(u) # admitted point symmetriesarbitrary 4 Y1 =??x , Y2 =??t , Y3 = x??x + 2t??t + v??v ,Y4 = ??vu? (? , 0) 5 Y1, Y2, Y3, Y4, Y5 = x ??x + 2?u ??u +(1 + 2?)v ??veu 5 Y1, Y2, Y3, Y4, Y6 = x ??x + 2??u + (2x + v) ??vu?2 ?Y1, Y2, Y3, Y4, Y5 (? = ?2) ,Y7 = ?xv ??x + (xu + v)u ??u + 2t ??v ,Y8 = ?x(2t + v2) ??x + 4t2 ??t + u(6t + 2xuv + v2) ??u+4tv ??v ,Y? = F(v, t) ??x ? u2G(v, t) ??u ,where (F(v, t),G(v, t)) is an arbitrary solutionof the linear system: Ft = Gv, Fv = G11+u2 e? arctan u 5 Y1, Y2, Y3, Y4,Y9 = v ??x + ?t??t ? (1 + u2) ??u ? x ??vMoreover, we present the point symmetry classification of the couplet potential503.3. Nonlocal symmetriessystem (3.19) in Table 3.3 [25], modulo its equivalence transformations?t = a28a4t + a1,x? = a?17 a8x + a5a?17 a8,u? = a27u + a6a27,v? = a6a7a8x + a7a8v + a2,w? = a28w + a28a5v +12a6a28x2 + a5a6a28x + a3,?K = a?14 a?27 K,(3.54)and?t = t,x? = x ? a9v,u? = u1 ? a9u,v? = v,w? = ?a92v2 + w,?K = (1 ? a9u)2K,(3.55)where a1, . . . , a9 are arbitrary constants with a4a7a8 , 0.Comparing Tables 3.1, 3.2 and 3.3, it is immediate to conclude that for somespecial cases of K(u), the potential system (3.5) and the couplet potential system(3.19) yield nonlocal symmetries of the nonlinear diffusion equation (3.4). Forexample, when K(u) = 11+u2 e? arctan u, Y9 and Z12 yield the same nonlocal symmetryof the nonlinear diffusion equation (3.4). In addition, the couplet potential system(3.19) could also yield nonlocal symmetries of the potential system (3.5). Forexample, the symmetry Z8 yields a nonlocal symmetry of the potential system (3.5)when K(u) = u? 43 ; the symmetry Z9 yields a nonlocal symmetry of the potentialsystem (3.5) when K(u) = u? 23 .Proposition 3.3.5 The symmetry X6 yields a nonlocal symmetry of the potentialsystem (3.5) with K(u) = u? 43 .Proof. Suppose the symmetry X6 yields a local symmetry of the potential system(3.5) with K(u) = u? 43 . Consequently, there must exist a differential function f [u, v]such that, in evolutionary form, ?X6 = (?3xu ? x2ux) ??u + f [u, v] ??v is a local sym-metry of the potential system (3.5). Since vx = u, vt = u? 43 ux and ut = (u? 43 ux)x,one can restrict f [u, v] to be of the form f (x, t, u, v, ux, uxx, ...) depending on x, t, u513.3. Nonlocal symmetriesTable 3.3: Point symmetry classification for the potential system (3.19)K(u) # admitted point symmetriesarbitrary 5 Z1 =??x + v??? , Z2 =??t , Z3 =??v , Z4 =??? ,Z5 = x ??x + 2t??t + v??v + 2????u? (? , 0) 6 Z1, Z2, Z3, Z4, Z5,Z6 = x ??x +2?u??u +(1 + 2?)v ??v + 2?(1 + 1?)???eu 6 Z1, Z2, Z3, Z4, Z5,Z7 = x ??x + 2??u + (2x + v) ??v + (x2 + 2?) ???u?43 7 Z1, Z2, Z3, Z4, Z5, Z6 (? = ?43 ),Z8 = x2 ??x ? 3xu??u ? ???vu?23 7 Z1, Z2, Z3, Z4, Z5, Z6 (? = ?23 ),Z9 = (xv ? ?) ??x ? 3uv ??u ? v2 ??v ? v? ???u?2 ?Z1, Z2, Z3, Z4, Z5, Z6 (? = ?2) ,Z10 = ?(xv + ?) ??x + (2xu + v)u ??u + 2t ??v?v? ??? ,Z11 = ?(6xt + xv2 + 2v?) ??x + 4t2 ??t+u(10t + 4xuv + 2u? + v2) ??u+4tv ??v ? (2t + v2)? ??? ,Z? = F(v, t) ??x ? u2G(v, t) ??u + H(v, t) ??v ,where (F(v, t),G(v, t),H(v, t)) is an arbitrarysolution of the linear system:Fv = G, Hv = F, Ht = G11+u2 e? arctan u6Z1, Z2, Z3, Z4, Z5,Z12 = v ??x + ?t??t ? (1 + u2) ??u ? x ??v + v2?x22???523.4. Nonlocally related systems in three or more dimensionsand the partial derivatives of u with respect to x. Firstly, suppose f [u, v] is of theform f (x, t, u, v, ux). Applying ?X(?)6 to the potential system (3.5), one obtainsfx + fuux + fvvx + fux uxx = ?3xu ? x2ux,ft + fuut + fvvt + fux utx = 43 (3xu + x2ux)u?73 ux + Dx(?3xu ? x2ux)u? 43 (3.56)on every solution of the potential system (3.5). After making appropriate substitu-tions and equating the coefficients of the term uxx, one obtains fux = 0. By simi-lar reasoning, one can show that f (x, t, u, v, ux, uxx, ...) has no dependence on anypartial derivative of u with respect to x. Hence f [u, v] is of the form f (x, t, u, v).Consequently, if X6 yields a local symmetry of the nonlinear diffusion equation(4.45) with K(u) = u? 43 , then ?X6 must be a point symmetry of the correspondingpotential system (3.5).Comparing Tables 3.1 and 3.2, one immediately sees that symmetry X6 doesnot yield a point symmetry of the corresponding potential system (3.5). This fol-lows from the fact that when K(u) = u? 43 , the potential system (3.5) has no pointsymmetry whose infinitesimal components corresponding to the variables (x, t) arethe same as those for X6. Hence X6 yields a nonlocal symmetry of the potentialsystem (3.5) with K(u) = u? 43 .Remark 3.3.6 Proposition 3.3.5 shows that a local symmetry of a subsystem of agiven PDE system can yield a nonlocal symmetry of the given PDE system, sincethe point symmetry X6 of the nonlinear diffusion equation (3.4), as a subsystem of(3.5), yields a nonlocal symmetry of the potential system (3.5).3.4 Nonlocally related systems in three or moredimensionsIn previous sections we showed how to construct nonlocally related PDE systemsfor a PDE system with two independent variables, and how to use such nonlocallyrelated PDE systems to find nonlocal symmetries and nonlocal CLs for the givenPDE system. Now consider a PDE system R{x; u} with n (n ? 3) independentvariables x = (x1, . . . , xn) and m dependent variables u = (u1, . . . , um), given byR?[u] = 0, ? = 1, . . . , s. (3.57)The situation in the case of n ? 3 independent variables is more complicatedthan in the case of two independent variables, since there exist several differenttypes of CLs in higher dimensions. In this section, we present a systematic methodfor the construction of nonlocally related PDE systems of R{x; u} (3.57) using533.4. Nonlocally related systems in three or more dimensionsdivergence-type CLs [7, 25, 43, 44]. For nonlocally related PDE systems arisingfrom lower-degree CLs, one can refer to [25, 43, 44] for more details.Suppose R{x; u} (3.57) has a local CL given bydiv(?[u]) =n?i=1Di?i[u] = 0. (3.58)By introducing n2 potential variables v jk ( j, k = 1, . . . , n) with v jk = ?vk j, oneobtains n potential equationsn?j=1D jvi j = ?i[u], i = 1, . . . , n. (3.59)Since v jk = ?vk j, the potential equations (3.59) only involve n(n?1)2 potential vari-ables, say v jk ( j < k). It is straightforward to show that the system of potentialequations (3.59) is equivalent to the CL (3.58). Note that the system of potentialequations (3.59) is under-determined. In particular, the system of potential equa-tions (3.59) is invariant under the transformationsvi j ? vi j + Dkwi jk, (3.60)where wi jk are n(n?1)(n?2)6 arbitrary functions that are components of a totally an-tisymmetric tensor. Hence the system of potential equations (3.59) has an infinitenumber of point symmetriesXgauge =?i, j,kDkwi jk??vi j, (3.61)which are called gauge symmetries.The system of potential equations (3.59) together with the given PDE systemR{x; u} (3.57) yields a potential system S{x; u, v} of R{x; u}, given byR?[u] = 0, ? = 1, . . . , s.n?j=1D jvi j = ?i[u], i = 1, . . . , n. (3.62)As in the case of two dependent variables, one can show that the potentialsystem S{x; u, v} (3.62) is nonlocally related to R{x; u} (3.57).Example 3.4.1 Consider a PDE system R{x, y, z; u} with three independent vari-ables. Suppose R{x, y, z; u} has a local CLdiv(?i[u]) = 0. (3.63)543.4. Nonlocally related systems in three or more dimensionsHence one can introduce three potential variables v = (v1, v2, v3) to obtain threepotential equationsv3y ? v2z = ?1[u],v1z ? v3x = ?2[u],v2x ? v1y = ?3[u].(3.64)Therefore, a potential system S{x, y, z; u, v} of R{x, y, z; u} is given by R{x, y, z; u}and the system of potential equations (3.64). The potential system S{x, y, z; u, v} isinvariant under the transformations(v1, v2, v3) ? (v1, v2, v3) + (Dxw[u],Dyw[u],Dzw[u]). (3.65)It follows that the gauge symmetries of S{x, y, z; u, v} are given byXgauge = Dxw[u]??v1+ Dyw[u]??v2+ Dzw[u]??v3. (3.66)Due to the under-determined property of the potential system, the situation forseeking nonlocal symmetries is different in three or more independent variablescase. The following important theorem shows nonlocal symmetries cannot arisefrom the under-determined potential system S{x; u, v} (3.62) [7, 25].Theorem 3.4.2 Each local symmetry of an under-determined potential system S{x;u, v} (3.62) projects onto a local symmetry of R{x; u} (3.57).In order to eliminate the gauge freedom of the potential system S{x; u, v} (3.62),it is necessary to add gauge constraints to S{x; u, v}. The choice of gauge constraintswill depend on particular problems. But the corresponding gauge-constrained(determined) potential system ?S{x; u, v} must have the property: all solutions ofS{x; u, v} can be obtained from the solutions of ?S{x; u, v}. Examples of gauge con-straints for the potential system (3.64):? divergence gauge: div(v) = v1x + v2y + v3z = 0,? algebraic gauge: vk = 0, k = 1 or 2 or 3,? Poincare? gauge: xv1 + yv2 + zv3 = 0.If x represents the time, i.e, x = t, the following gauge constraints are fre-quently used in applications:? Lorentz gauge: v1t ? v2y ? v3z = 0,? Cronstrom gauge: tv1 ? yv2 ? zv3 = 0.553.5. Relationships between local symmetries of PDE systemsIt is shown that a determined potential system could yield nonlocal symmetriesof a given PDE system with three or more independent variables [7, 25, 43, 44].However, not all determined potential systems can yield nonlocal symmetries of agiven PDE system. For example, consider the wave equationutt = uxx + uyy. (3.67)A determined potential system is obtained by adding the Lorentz gauge to its under-determined potential system. Hence the determined potential system is given byv3x ? v2y = ut,v1y ? v3t = ?ux,v2t ? v1x = ?uy,v1t ? v2y ? v3z = 0.(3.68)It turns out that there exist point symmetries of the potential system (3.68) thatproject onto nonlocal symmetries of the wave equation (3.67) [25, 44]. However,it is also shown that if one replaces the Lorentz gauge by the divergence gauge, thealgebraic gauge, the Poincare? gauge, or the Cronstrom gauge, no nonlocal sym-metries of the wave equation (3.67) arise from these determined potential systems[25, 44].Similar to the case of two independent variables, one can construct a tree ofnonlocally related PDE systems of R{x; u} (3.57). In seeking additional CLs forthe potential systems, there is a similar result to Theorem 3.2.20 [25, 26, 63].Theorem 3.4.3 Suppose R{x; u} (3.57) has a CL (3.58). Let S{x; u, v} be the po-tential system of R{x; u} consisting of R{x; u} and the potential equations (3.59).Then each CL of S{x; u, v}, arising from multipliers that do not involve the potentialvariables v, is equivalent to a local CL of R{x; u}.It is important to note that Theorem 3.4.3 does not hold for a potential systemwith a gauge constraint [25]. Moreover, unlike the situation for nonlocal symme-tries, nonlocal CLs can arise from both determined and under-determined potentialsystems [11, 25].3.5 Relationships between local symmetries of PDEsystemsAs discussed in Section 3.3, an effective way to seek nonlocal symmetries of agiven PDE system is to apply Lie?s algorithm to PDE systems in a tree of nonlocally563.5. Relationships between local symmetries of PDE systemsrelated PDE systems. It has been shown that both potential systems and subsystemscan yield nonlocal symmetries. Does there exist any relationship between localsymmetries of a given PDE system and those of its potential systems?In next new theorem, a correspondence between local symmetries of a givenPDE system having precisely n linearly independent local CLs and those of itspotential systems is presented.Theorem 3.5.1 Suppose a PDE system R{x, t; u} with two independent variables(x, t) and m dependent variables u = (u1, . . . , um) given byR?[u] = 0, ? = 1, . . . , s, (3.69)has precisely n linearly independent local CLs. Then any local symmetry of thePDE system R{x, t; u} (3.69) can be obtained by projection of some local symmetryof its n-plet potential system.Proof. Let the n local CLs of R{x, t; u} (3.69) be given by Dt? j[u]+Dx(?? j[u])=0 for some densities ? j[u] and fluxes ?? j[u], j = 1, . . . , n. Then the correspondingn-plet potential system of R{x, t; u} (3.69) is given byvjx = ? j[u],vjt = ? j[u], j = 1, . . . , n,R?[u] = 0, ? = 1, . . . , s.(3.70)Suppose ?X =m?i=1?i[u] ??uiis a local symmetry of R{x, t; u} (3.69). It sufficesto prove that there exist functions ? j[u, v], j = 1, . . . , n, such that ?Y = ?X +n?j=1? j[u, v] ??v jis a local symmetry of the n-plet potential system (3.70).Applying the corresponding infinite prolongation?Y(?) = ?X(?) +n?j=1? j[U,V] ??V j+?JDJ? j[U,V]??V jJto the functionsV jx ? ? j[U],V jt ? ? j[U], j = 1, . . . , n,R?[U], ? = 1, . . . , s,(3.71)573.5. Relationships between local symmetries of PDE systemsone obtainsDx? j[U,V] ? ?X(?)? j[U],Dt? j[U,V] ? ?X(?)? j[U], j = 1, . . . , n,?X(?)R?[U], ? = 1, . . . , s.(3.72)Then ?Y is a local symmetry of the n-plet potential system (3.70) if and only if (3.72)vanishes on any solution (U,V) = (u, v) = (u(x, t), v(x, t)) of the n-plet potentialsystem (3.70). From Theorem 2.2.32, ?X is a local symmetry of R{x, t; u} (3.69) ifand only if its infinite prolongation ?X(?) satisfies?X(?)R?[u] = 0, ? = 1, . . . , s, (3.73)on any solution of R{x, t; u} (3.69). Hence the equations (3.73) hold on any solutionof the n-plet potential system (3.70). Therefore, it suffices to prove that there existsome functions ? j[u, v] so that the equationsDx? j[u, v] = ?X(?)? j[u],Dt? j[u, v] = ?X(?)? j[u], j = 1, . . . , n,(3.74)hold on any solution of the n-plet potential system (3.70).Theorem 2.3.16 shows that a symmetry maps fluxes (densities) to fluxes (den-sities). Since ?X is a local symmetry of R{x, t; u} (3.69), the entries ?? j[u] =?X(?)? j[u] and ? ?? j[u] = ?X(?)(?? j[u]), j = 1, . . . , n, must be densities and fluxesof CLs of R{x, t; u} (3.69), i.e., for each j = 1, . . . , n, Dt ?? j[u]+Dx(? ?? j[u])= 0 isa CL of R{x, t; u} (3.69). Since R{x, t; u} (3.69) has precisely n linear independentlocal CLs, it follows that, for each j = 1, . . . , n,?X(?)? j[u] = ?? j[u] =n?k=1ajk?k[u] + T j[u],?X(?)? j[u] = ?? j[u] =n?k=1ajk?k[u] + S j[u],(3.75)for some constants a jk, k = 1, . . . , n, and DtTj[u] ? DxS j[u] = 0 is a trivial CL ofR{x, t; u} (3.69). In particular, for each j = 1, . . . , n, T j[u] = A j[u] + B j[u] andS j[u] = F j[u] + G j[u], where DtA j[u] ? DxF j[u] = 0 is a first kind trivial CL andDtB j[u] ? DxG j[u] = 0 is a second kind trivial CL. It follows that (A j[u], F j[u])vanish on any solution of R{x, t; u} (3.69), and hence (A j[u], F j[u]) vanish on anysolution of the n-plet potential system (3.70). Since (B j[u],G j[u]) yield a null583.5. Relationships between local symmetries of PDE systemsdivergence, according to Theorem 2.3.3, there exists some function H j[u] suchthat B j[u] = DxH j[u] and G j[u] = DtH j[u].Now let ? j[U,V] be the functions? j[U,V] =n?k=1ajkVk + H j[U], j = 1, . . . , n. (3.76)Then, on any solution (U,V) = (u, v) = (u(x, t), v(x, t)) of the n-plet potential sys-tem (3.70), one hasDx? j[u, v] =n?k=1ajkvkx + DxH j[u] =n?k=1ajk?k[u] + B j[u]=n?k=1ajk?k[u] + B j[u] + A j[u] =n?k=1ajk?k[u] + T j[u]= ?X(?)? j[u],Dt? j[u, v] =n?k=1ajkvkt + DtH j[u] =n?k=1ajk?k[u] + G j[u]=n?k=1ajk?k[u] + G j[u] + F j[u] =n?k=1ajk?k[u] + S j[u]= ?X(?)? j[u], j = 1, . . . , n.(3.77)Hence, ? j[u, v], j = 1, . . . , n, given by (3.76) when (U,V) = (u, v) is a solution ofthe n-plet potential system (3.70), satisfy the equations (3.74).By construction,?Y = ?X +n?j=1????????n?k=1ajkvk + H j[u]??????????v jis a local symmetry of the n-plet potential system (3.70), whose projection is thelocal symmetry ?X =m?i=1?i[u] ??uiof the given PDE system R{x, t; u} (3.69). Remark 3.5.2 The proof of Theorem 3.5.1 shows how to directly construct thelocal symmetry ?Y of the n-plet potential system (3.70) for any local symmetry ?Xof a given PDE system R{x, t; u} (3.69) which has precisely n local CLs.Corollary 3.5.3 Consider a PDE system R{x, t; u} with two independent variables(x, t) and m dependent variables u = (u1, . . . , um) given byR?[u] = 0, ? = 1, . . . , s. (3.78)593.5. Relationships between local symmetries of PDE systemsSuppose ?X is a local symmetry in evolutionary form of R{x, t; u} and Dt?i[u] ?Dx?i[u] = 0, i = 1, . . . , k, are linearly independent local CLs of R{x, t; u} (3.78).If for each ? = 1, . . . , k, the CLDt(?X(?)??[u])? Dx(?X(?)??[u])= 0is equivalent to the CLDt????????k?i=1a?i ?i[u]???????? ? Dx????????k?i=1a?i ?i[u]???????? = 0,for some constants a?i , i = 1, . . . , k, then ?X can be obtained by projection of somelocal symmetry of the corresponding k-plet potential system given byvix = ?i[u],vit = ?i[u], i = 1, . . . , k,R?[u] = 0, ? = 1, . . . , s.(3.79)Proof. The proof in Theorem 3.5.1 can be directly extended to the k-plet potentialsystem (3.79). Remark 3.5.4 Both Theorem 3.5.1 and Corollary 3.5.3 hold for PDE systems withthree or more independent variables, since one can directly extend the above proofsto such PDE systems.Example 3.5.5 Consider the nonlinear diffusion equationut =(u?43 ux)x. (3.80)From Tables 3.1 and 3.3, one sees that all point symmetries of the nonlinear diffu-sion equation (3.80) can be obtained by projection of some point symmetry of its2-plet potential system (3.19). For this example, we illustrate how to use Theorem3.5.1 to obtain this conclusion.Consider the point symmetry X = ?x2 ??x + 3xu??u of the nonlinear diffusionequation (3.80), whose evolutionary form is given by ?X = (3xu + x2ux) ??u . Usingthe direct method and Theorem 2.3.13, one can show that the nonlinear diffusionequation (3.80) has exactly two linearly independent local CLs given byut ?(u?43 ux)x= 0, (3.81)603.5. Relationships between local symmetries of PDE systems(xu)t ?(xu?43 ux + 3u?13)x= 0. (3.82)From Theorem 3.5.1, the point symmetry ?X of the nonlinear diffusion equation(3.80) can be obtained by projection of some local symmetry ?Y of its 2-plet poten-tial system given byvx = u,vt = u?43 ux,?x = xu,?t = xu?43 ux + 3u?13 .(3.83)Here ?1[u] = u, ?1[u] = u? 43 ux, ?2[u] = xu, ?2[u] = xu? 43 ux+3u? 13 . In particular,one can explicitly find ?Y without applying the Lie?s algorithm to the 2-plet potentialsystem (3.83). Applying the corresponding infinite prolongation ?X(?) to the fluxesof the CL (3.81), one obtains?X(?)(?1[u])= ?X(?)(u) = 3xu + x2ux = xu + x2ux + 2xu= ?2[u] + T 1[u],?X(?)(?1[u])= ?X(?)(u? 43 ux) = 3u?13 + xu?43 ux ? 43 x2u?73 u2x + x2u?43 uxx,= ?2[u] + S 1[u],(3.84)where DtT 1[u] ? DxS 1[u] = 0 is the trivial CL given byDt(x2ux + 2xu) ? Dx(? 43 x2u?73 u2x + x2u?43 uxx ? x2ut + x2ut)= Dt(Dx(x2u)) ? Dx(x2(? 43u? 73 u2x + x2u?43 uxx ? ut)+ Dt(x2u))= 0.(3.85)Applying the corresponding infinite prolongation ?X(?) to the fluxes of the CL(3.82), one obtains?X(?)(?2[u])= ?X(?)(xu) = 3x2u + x3ux = T 2[u],?X(?)(?2[u])= ?X(?)(xu? 43 ux + 3u? 13 ) = x3u? 43 uxx ? 43 x3u?73 u2x = S 2[u],(3.86)where DtT 2[u] ? DxS 2[u] = 0 is the trivial CL given byDt(Dx(x3u)) ? Dx(Dt(x3u) + x3(u?43 uxx ?43u? 73 u2x ? ut))= 0. (3.87)It follows that the constants and the functions H j[u], j = 1, 2, in (3.76) are givenbya11 = 0, a12 = 1, a21 = 0, a22 = 0, H1[u] = x2u, H2[u] = x3u.613.6. SummaryConsequently, ?Y = ?X + (? + x2u) ??v + x3u ??? = ?X + (? + x2vx) ??v + x2?x ??? , whichis the evolutionary form for the point symmetry Y = X + ? ??v .Besides being useful for obtaining nonlocal CLs and nonlocal symmetries ofa given PDE system R{x; u}, nonlocally related CL systems of R{x; u} have vari-ous other important applications. For instance, one can use nonlocal symmetriesto construct new solutions, which are not invariant solutions of local symmetries,of R{x; u} arising from invariant solutions of its nonlocally related PDE systems[41]. In [37], it was shown that new solutions can also arise from the ?nonclas-sical symmetries? of nonlocally related PDE systems of R{x; u}. In addition, onecan construct nonlocal mappings that linearize a given PDE system through itsnonlocally related CL systems [25]. The well-known Hopf-Cole transformation(see [52]) that linearizes Burgers? equation can be obtained through its potentialsystem [59, 90]. One can also construct a nonlocal mapping that maps a scalarPDE with variable coefficients to a linear PDE with constant coefficients throughits nonlocally related CL systems [33, 34]. Moreover, in [5], Anco and Blumanused nonlocal symmetries to obtain CLs of a given PDE.3.6 SummaryIn this chapter, we presented the known CL-based method for constructing nonlo-cally related CL systems of a given PDE system. Such nonlocally related CL sys-tems are important in obtaining nonlocal CLs, nonlocal symmetries and new exactsolutions of a given PDE system. Besides the known results, we introduced twonew results in this chapter. Theorem 3.2.7 showed that for two potential systemswritten in Cauchy-Kovalevskaya form, arising from two nontrivial and linearly in-dependent local CLs, the potential variable in one system is a nonlocal variable ofthe other system. Theorem 3.5.1 showed that any local symmetry of a PDE systemhaving precisely n local CLs must be a projection of some local symmetry of itscorresponding n-plet potential system.62Chapter 4Symmetry-based Method forConstructing Nonlocally RelatedPDE Systems4.1 IntroductionIn Chapter 3, we presented the CL-based method for constructing nonlocally re-lated CL systems of a given PDE system and the method for constructing subsys-tems. Moreover, some important applications of nonlocally related PDE systems,such as using nonlocally related PDE systems to find nonlocal symmetries andnonlocal CLs, were presented.In the CL-based method, to construct nonlocally related PDE systems of agiven scalar PDE, it is necessary that the given PDE has at least one nontrivialCL. A natural problem is how to construct nonlocally related PDE systems for ascalar PDE that has no nontrivial CL. For example, consider the nonlinear reaction-diffusion equationut ? uxx = Q(u), (4.1)where the reaction term Q(u) is an arbitrary constitutive function.According to Theorem 2.3.13, one can find all local CLs of the nonlinearreaction-diffusion equation (4.1) by the direct method. In particular, it sufficesto consider multipliers of the form ? = ?(x, t,U,Ux,Uxx,Uxxx). Applying the di-rect method, one obtains that ?(Ut ?Uxx ? Q(U)) is a divergence form if and onlyif ? satisfies the following equations:?U = 0, ?Ux = 0, ?Uxx = 0, ?Uxxx = 0,??tx + ??xxx ? ?x(?xx + ?t) = 0,??tt + ??txx ? ?t(?xx + ?t) = 0,?t + ?xx + ?Q?(U) = 0.(4.2)From equations (4.2), one immediately concludes that, for any nonlinear func-tion Q(u), the nonlinear reaction-diffusion equation (4.1) has no nontrivial local634.2. Nonlocally related PDE systems arising from point symmetriesCL. Consequently, it is impossible to construct a nonlocally related PDE system ofthe nonlinear reaction-diffusion equation (4.1) via the CL-based method.In this chapter, we introduce a new systematic method to construct nonlocallyrelated PDE systems which can be applied to a PDE system that has no nontrivialCL. In particular, we show that, for any given PDE system, a nonlocally relatedPDE system arises naturally from each point symmetry of the given PDE system.Consequently, one can extend a tree of nonlocally related PDE systems by addingthis method to Procedure 3.2.15 for the construction of a tree of nonlocally relatedPDE systems.Unlike the CL-based method, the symmetry-based method can be directly ap-plied to a PDE system with three or more independent variables. More importantly,nonlocal related PDE systems arising from the symmetry-based mathod are deter-mined. In addition, by various examples, it is shown that a nonlocally related PDEsystem arising from a point symmetry of a given PDE system could also yieldnonlocal symmetries of the given PDE system.4.2 Nonlocally related PDE systems arising from pointsymmetriesConsider a PDE system R{x, t; u} of order l with two independent variables (x, t)and m dependent variables u = (u1, . . . , um) given byR?[u] = R?(x, t, u, ?u, ?2u, . . . , ?lu) = 0, ? = 1, . . . , s. (4.3)Suppose the PDE system R{x, t; u} (4.3) has a point symmetry with infinitesi-mal generator X = ?(x, t, u) ??x + ?(x, t, u)??t +m?i=1?i(x, t, u) ??ui. By introducing thecanonical coordinates corresponding to X:X = X(x, t, u),T = T (x, t, u),U i = U i(x, t, u), i = 1, . . . ,m,(4.4)satisfyingXX = 0,XT = 0,XU1 = 1,XU i = 0 i = 2, . . . ,m,(4.5)644.2. Nonlocally related PDE systems arising from point symmetriesone maps X into the canonical form Y = ??U1 while the PDE system R{x, t; u} (4.3)becomes the invertibly equivalent PDE system ?R{X, T ; U} in terms of the canoni-cal coordinates (X, T,U). Since an invertible transformation maps a symmetry of aPDE system to a symmetry of the transformed system, Y is the infinitesimal gen-erator of a point symmetry of ?R{X, T ; U}. Consequently, ?R{X, T ; U} is invariantunder translations in U1. It follows that ?R{X, T ; U} is of the form?R?(X, T, ?U, ?U, . . . , ?lU) = 0, ? = 1, . . . , s, (4.6)where ?U = (U2, . . . ,Um).Introducing two new variables ? and ?, related to the first partial derivatives ofU1, one obtains the equivalent PDE system ?R{X, T ; U, ?, ?} given by? = U1T ,? = U1X,?R?(X, T, ?U, ?, ?, ? ?U, . . . , ?l?1?, ?l?1?, ?l ?U) = 0, ? = 1, . . . , s,(4.7)where ?R?(X, T, ?U, ?, ?, ? ?U, . . . , ?l?1?, ?l?1?, ?l ?U) = 0 is obtained from ?R?(X, T, ?U,?U, . . . , ?lU) = 0 after making the appropriate substitutions. The PDE system?R{X, T ; U, ?, ?} (4.7) is called the intermediate system of ?R{X, T ; U} (4.6).According to Theorem 3.2.14, the PDE system ?R{X, T ; U, ?, ?} (4.7) is locallyrelated to the PDE system ?R{X, T ; U} (4.6), and hence locally related to the givenPDE system R{x, t; u} (4.3).Excluding the dependent variable U1 from the PDE system ?R{X, T ; U, ?, ?}(4.7), one obtains the equivalent PDE system ?R{X, T ; ?U, ?, ?}?X = ?T ,?R?(X, T, ?U, ?, ?, ? ?U, . . . , ?l?1?, ?l?1?, ?l ?U) = 0, ? = 1, . . . , s. (4.8)According to Theorem 3.2.14, since the PDE system ?R{X, T ; ?U, ?, ?} (4.8)is obtained by excluding U1 from the PDE system ?R{X, T ; U, ?, ?} (4.7), whichcannot be expressed as a local function of ?, ? and their derivatives, it followsthat the PDE system ?R{X, T ; ?U, ?, ?} (4.8) is nonlocally related to the PDE sys-tem ?R{X, T ; U, ?, ?} (4.7). In particular, if (?, ?,U2, . . . ,Um) = ( f (x, t), g(x, t),h2(x, t), . . . , hm(x, t)) solves the PDE system ?R{X, T ; ?U, ?, ?} (4.8), there exists aspectrum of functions U1 = h1(x, t)+C, where C is an arbitrary constant, such that(?, ?,U1,U2, . . . ,Um) = ( f (x, t), g(x, t), h1(x, t) + C, h2(x, t), . . . , hm(x, t)) solvesthe intermediate system ?R{X, T ; U, ?, ?} (4.7). By projection, (U1,U2, . . . ,Um) =(h1(x, t)+C, h2(x, t), . . . , hm(x, t)) is a solution of the PDE system ?R{X, T ; U} (4.6).Thus the correspondence between the solutions of the PDE system ?R{X, T ; ?U, ?, ?}654.2. Nonlocally related PDE systems arising from point symmetries(4.8) and those of the PDE system ?R{X, T ; U} (4.6) is not one-to-one. It followsthat the PDE system ?R{X, T ; ?U, ?, ?} (4.8) is nonlocally related to the PDE system?R{X, T ; U} (4.6), and hence nonlocally related to the given PDE system R{x, t; u}(4.3).Since the way one constructs the PDE system ?R{X, T ; ?U, ?, ?} (4.8) is in the re-verse direction of the construction for a potential system, we call the PDE system?R{X, T ; ?U, ?, ?} (4.8) an inverse potential system. Since the inverse potential sys-tem arising from a point symmetry of a given PDE system is nonlocally related tothe given PDE system, we use the terminology nonlocally related symmetry-basedsystem to denote an inverse potential system.Based on the above discussions, one has proved the following theorem.Theorem 4.2.1 Any point symmetry of a given PDE system R{x, t; u} (4.3) yieldsa nonlocally related PDE system (inverse potential system) of the given PDE sys-tem R{x, t; u} (4.3) given by the PDE system ?R{X, T ; ?U, ?, ?} (4.8).Remark 4.2.2 Connection between the symmetry-based method and the CL-basedmethod. The symmetry-based method to obtain a nonlocally related PDE systemdoes not require the existence of a nontrivial local CL of a given PDE system. Thusthe new method is complementary to the CL-based method for constructing nonlo-cally related PDE systems. In particular, for the CL-based method, the constructedsystem is a potential system of the given PDE system. For the symmetry-basedmethod, since the intermediate system is locally related to the given PDE system,one can treat the intermediate system as the starting PDE system. In this sense,for the symmetry-based method, the starting PDE system is a potential systemof the final constructed system (the inverse potential system). It follows that thesymmetry-based method is in the reverse direction of the CL-based method.Remark 4.2.3 The situation for a PDE system with at least three independent vari-ables. The symmetry-based method can be adapted to a PDE system which has atleast three independent variables. Without loss of generality, consider a scalar PDER{x; u} with n ? 3 independent variables x = (x1, . . . , xn) and one dependent vari-able u:R(x, u, ?u, ?2u, . . . , ?lu) = 0. (4.9)Suppose the scalar PDE R{x; u} (4.9) has a point symmetry with the infinitesimalgenerator X. The canonical coordinates corresponding to X:Xi = Xi(x, u), i = 1, . . . , n,U = U(x, t, u), (4.10)664.2. Nonlocally related PDE systems arising from point symmetriessatisfyingXXi = 0, i = 1, . . . , n,XU = 1,(4.11)maps X into the canonical form Y = ??U . In terms of (X, T,U) coordinates, thegiven scalar PDE R{x; u} (4.9) becomes the invertibly related PDE ?R{X; U} (X =(X1, . . . , Xn)) of the form?R(X, ?U, ?2U, . . . , ?lU) = 0. (4.12)Introducing the new variables ? = (?1, . . . , ?n), related to the first partialderivatives of U, one obtains the equivalent locally related intermediate system?R{X; U, ?} given by?i = UXi , i = 1, . . . , n,?R(X, ?, ?? . . . , ?l?1?) = 0, (4.13)where ?R(X, ?, ??, . . . , ?l?1?) = 0 is obtained from ?R(X, ?U, ?2U, . . . , ?lU) = 0after making the appropriate substitutions. Excluding U from the PDE system?R{X; U, ?} (4.13), one obtains the inverse potential system ?R{X;?}?iX j ? ?jXi = 0, i, j = 1, . . . , n,?R(X, ?, ??, . . . , ?l?1?) = 0.(4.14)By construction, one can show that the inverse potential system ?R{X;?} (4.14) isnonlocally related to the scalar PDE ?R{X; U} (4.12), hence nonlocally related tothe scalar PDE R{x; u} (4.9). Moreover, since the inverse potential system ?R{X;?}(4.14) has curl-type CLs, it could possibly yield nonlocal symmetries of the scalarPDE R{x; u} (4.9) from local symmetries of the inverse potential system ?R{X;?}(4.14) [25, 43, 44].Corollary 4.2.4 Consider an evolutionary scalar PDE R{x, t; u}, invariant under-translations in u, given byut = F(x, t, ?xu, . . . , ?lxu). (4.15)Let ? = ?xu = ux. Then the scalar PDE?t = DxF(x, t, ?, ?x?, . . . , ?l?1x ?) (4.16)is a locally related subsystem of an inverse potential system of the PDE R{x, t; u}(4.15).674.3. Examples of inverse potential systems arising from point symmetriesProof. Introducing the new variables ? and ?, related to the first partial derivativesof u, one obtains the following locally related intermediate system ?R{x, t; u, ?, ?}of the given PDE R{x, t; u} (4.15):? = ut,? = ux,? = F(x, t, ?, ?x?, . . . , ?l?1x ?).(4.17)Excluding the dependent variable u from the intermediate system ?R{x, t; u, ?, ?}(4.17), one obtains the inverse potential system ?R{X, T ; ?U, ?, ?}?x = ?t,? = F(x, t, ?, ?x?, . . . , ?l?1x ?).(4.18)From the previous discussion, the PDE system ?R{X, T ; ?U, ?, ?} (4.18) is nonlocallyrelated to (4.17). Furthermore, one can exclude the dependent variable ? from thePDE system ?R{X, T ; ?U, ?, ?} (4.18) to obtain the scalar PDE ?R{x, t; ?}?t = DxF(x, t, ?, ?x?, . . . , ?l?1x ?). (4.19)Since the excluded variable ? can be expressed from the equations of the PDEsystem ?R{X, T ; ?U, ?, ?} (4.18) in terms of ? and its derivatives, the PDE ?R{x, t; ?}(4.19) is locally related to the inverse potential system ?R{X, T ; ?U, ?, ?} (4.18). 4.3 Examples of inverse potential systems arising frompoint symmetriesIn the previous section we introduced a new systematic symmetry-based methodto construct nonlocally related PDE systems (inverse potential systems) of a givenPDE system. Such equivalent PDE systems are nonlocally related to the given PDEsystem. In this section, we illustrate this method by several examples.4.3.1 Nonlinear reaction-diffusion equationConsider the nonlinear reaction-diffusion equation (4.1). As stated in Section 4.1,the nonlinear reaction-diffusion equation (4.1) has no local CL for any nonlinearterm Q(u). Thus it is impossible to construct nonlocally related PDE systems ofthe nonlinear reaction-diffusion equation (4.1) by the CL-based method.684.3. Examples of inverse potential systems arising from point symmetriesIn contrast, the nonlinear reaction-diffusion equation (4.1) has point symme-tries. Thus one can construct nonlocally related PDE systems of the the nonlin-ear reaction-diffusion equation (4.1) through the symmetry-based method intro-duced in Section 4.2. The point symmetry classification of the nonlinear reaction-diffusion (4.1) is presented in [47, 49] and exhibited in Table 4.1, modulo the groupof equivalent transformations (2.71).Table 4.1: Point symmetry classification for the reaction-diffusion equation (4.1)Q(u) # admitted point symmetriesarbitrary 2 X1 = ??x , X2 =??tua(a , 0, 1) 3 X1, X2, X3 = u ??u ? (a ? 1)t ??t ? a?12 x ??xeu 3 X1, X2, X4 = ??u ? t??t ?12 x??xu ln u 4 X1, X2, X5 = uet ??u ,X6 = 2et ??x ? xuet ??u(I) The case when Q(u) is arbitraryFor arbitrary Q(u), the nonlinear reaction-diffusion equation (4.1) has the exhibitedtwo point symmetries: X1 and X2. Therefore, using the symmetry-based methodone can use interchanges of x and u and also t and u to construct two inversepotential systems of the nonlinear reaction-diffusion equation (4.1).(I-a) Inverse potential system arising from X1After an interchange of the variables x and u, the nonlinear reaction-diffusion equa-tion (4.1) becomes the invertibly related PDE given byxt =xuu ? Q(u)x3ux2u. (4.20)Corresponding to the invariance of PDE (4.20) under translations of its depen-dent variable x, one obtains the following locally related intermediate system forthe nonlinear reaction-diffusion equation (4.1) by introducing two new variables:v = xu,w = xt,w = vu ? Q(u)v3v2.(4.21)Excluding x from the intermediate system (4.21), one obtains the inverse potential694.3. Examples of inverse potential systems arising from point symmetriessystem of the PDE system (4.21) given byvt = wu,w = vu ? Q(u)v3v2.(4.22)In addition, one can exclude w from the PDE system (4.22) to get the scalar PDEvt =(vu ? Q(u)v3v2)u. (4.23)By construction, the scalar PDE (4.23) is a locally related subsystem of the PDEsystem (4.22). Moreover, since the scalar PDE (4.23) is in a CL form and the non-linear reaction-diffusion equation (4.1) has no local CL, from Remark 2.3.15, it fol-lows that there is no invertible transformation that relates the scalar PDE (4.23) andthe nonlinear reaction-diffusion equation (4.1). Therefore, the scalar PDE (4.23) isnonlocally related to the nonlinear reaction-diffusion equation (4.1).(I-b) Inverse potential system arising from X2After an interchange of the variables t and u, the nonlinear reaction-diffusion equa-tion (4.1) becomest2u ? Q(u)t3u + t2utxx ? 2txtutxu + t2xtuu = 0, (4.24)which is not in solved form and has mixed derivatives.Corresponding to the invariance of PDE (4.24) under translations of its depen-dent variable t, one introduces two new variables ? = tx and ? = tu to obtainthe locally related intermediate system of the nonlinear reaction-diffusion equation(4.1) given by? = tx,? = tu,?2 ? Q(u)?3 + ?2?x ? 2???u + ?2?u = 0.(4.25)Excluding t from the intermediate system (4.25), one obtains another inverse po-tential system of the nonlinear reaction-diffusion equation (4.1) given by?u ? ?x = 0,?2 ? Q(u)?3 + ?2?x ? 2???u + ?2?u = 0,(4.26)which is nonlocally related to the nonlinear reaction-diffusion equation (4.1).The constructed inverse potential systems for the nonlinear reaction-diffusionequation (4.1) (Q(u) is arbitrary) are illustrated in Figure 4.1.704.3. Examples of inverse potential systems arising from point symmetries(4.1)(4.22), (4.23) (4.26)Figure 4.1: The constructed inverse potential systems for the nonlinear reaction-diffusion equation (4.1) (Q(u) is arbitrary), with the arrows pointing to the inversepotential systems.(II) Inverse potential system arising from X3 when Q(u) = u3When Q(u) = ua, (a , 0, 1), the nonlinear reaction-diffusion equation (4.1) has oneadditional point symmetry X3. For simplicity, we consider the case when a = 3,i.e., Q(u) = u3. Canonical coordinates induced by X3 are given byX = xu,T = tx2,U = ? ln x.(4.27)In (X, T,U) coordinates, the corresponding nonlinear reaction-diffusion equation(4.1) becomes the invertibly related PDE? 3U2X ? 2XU3X ? X3U3X ? U2XUT + 10TU2XUT + UXX ? 4TUT UXX+ 4T 2U2T UXX + 4T2U2XUTT + 4TUXUT X ? 8T 2UXUT UT X = 0.(4.28)Accordingly, introducing the new variables ? = UX and ? = UT , one obtainsthe locally related intermediate system of the nonlinear reaction-diffusion equation(4.1) given by? = UX,? = UT ,? 3?2 ? 2X?3 ? X3?3 ? ?2? + 10T?2? + ?X ? 4T??X+ 4T 2?2?X + 4T 2?2?T + 4T??X ? 8T 2???X = 0.(4.29)Excluding U from the intermediate system (4.29), one obtains an additional inversepotential system of the corresponding nonlinear reaction-diffusion equation (4.1)714.3. Examples of inverse potential systems arising from point symmetriesgiven by?T = ?X,? 3?2 ? 2X?3 ? X3?3 ? ?2? + 10T?2? + ?X ? 4T??X+ 4T 2?2?X + 4T 2?2?T + 4T??X ? 8T 2???X = 0,(4.30)which is nonlocally related to the nonlinear reaction-diffusion equation (4.1). More-over, comparing the number of point symmetries of the PDE system (4.29) and thePDE system (4.26), one is able to show there is no invertible transformation relat-ing these two systems. Hence, the PDE system (4.29) is nonlocally related to thePDE system (4.26).The constructed inverse potential systems for the nonlinear reaction-diffusionequation (4.1) (Q(u) = u3) are illustrated in Figure 4.2.(4.1)(4.30)(4.22), (4.23) (4.26)Figure 4.2: The constructed inverse potential systems for the nonlinear reaction-diffusion equation (4.1) (Q(u) = u3), with the arrows pointing to the inverse poten-tial systems.(III) Inverse potential system arising from X4 when Q(u) = euWhen Q(u) = eu, the nonlinear reaction-diffusion equation (4.1) admits one addi-tional point symmetry X4. Canonical coordinates induced by X4 are given byX = u + 2 ln x,T = tx2,U = ?2 ln x.(4.31)In (X, T,U) coordinates, the corresponding nonlinear reaction-diffusion equation(4.1) becomes the invertibly related PDE? 2U2X ? 2U3X ? eXU3X ? U2XUT + 6TUT U2X + 4UXX ? 8TUT UXX+ 4T 2U2T UXX + 4T2U2XUTT + 8TUXUT X ? 8T2UXUT UT X = 0.(4.32)724.3. Examples of inverse potential systems arising from point symmetriesIt follows that the introduction of the new variables ? = UX and ? = UT yieldsthe locally related intermediate system of the nonlinear reaction-diffusion equation(4.1) given by? = UX,? = UT ,? 2?2 ? 2?3 ? eX?3 ? ?2? + 6T?2? + 4?X ? 8T??X+ 4T 2?2?X + 4T 2?2?T + 8T??X ? 8T 2???X = 0.(4.33)Excluding U from the intermediate system (4.33), one obtains a third inverse po-tential system of the corresponding nonlinear reaction-diffusion (4.1) given by?T = ?X ,? 2?2 ? 2?3 ? eX?3 ? ?2? + 6T?2? + 4?X ? 8T??X+ 4T 2?2?X + 4T 2?2?T + 8T??X ? 8T 2???X = 0,(4.34)which is nonlocally related to the nonlinear reaction-diffusion equation (4.1). Sim-ilar to the situation in (II), one can show that the PDE system is nonlocally relatedto the PDE system (4.26).The constructed inverse potential systems for the nonlinear reaction-diffusionequation (4.1) (Q(u) = eu) are illustrated in Figure 4.3.(4.1)(4.22), (4.23) (4.26)(4.34)Figure 4.3: The constructed inverse potential systems for the nonlinear reaction-diffusion equation (4.1) (Q(u) = eu), with the arrows pointing to the inverse poten-tial systems.(IV) The case when Q(u) = u ln uWhen Q(u) = u ln u, the nonlinear reaction-diffusion equation (4.1) has two addi-tional point symmetries X5 and X6.(IV-a) Inverse potential system arising from X5734.3. Examples of inverse potential systems arising from point symmetriesCanonical coordinates induced by X5 are given byX = x,T = t,U = e?t ln u.(4.35)In (X, T,U) coordinates, the corresponding nonlinear reaction-diffusion equation(4.1) becomesUT = UXX + eT U2X. (4.36)Thus one introduces the new variables p = UX and q = UT to obtain the locallyrelated intermediate system of the nonlinear reaction-diffusion equation (4.1) givenbyp = UX,q = UT ,q = pX + eT p2.(4.37)Excluding U from the intermediate system (4.37), one obtains the inverse potentialsystem of the corresponding nonlinear reaction-diffusion (4.1) given bypT = qX ,q = pX + eT p2.(4.38)In addition, excluding q from the inverse potential system (4.37), one obtains thelocally related subsystem of the inverse potential system (4.38) given bypT = pXX + 2eT ppX , (4.39)which is in a CL form. The PDE (4.39) is in a CL form and the nonlinear reaction-diffusion equation (4.1) has no local CL. Hence the PDE (4.39) is nonlocally relatedto the nonlinear reaction-diffusion equation (4.1).(IV-b) Inverse potential system arising from X6Canonical coordinates induced by X6 are given byX = ex24 u,T = t,U = 12e?t x.(4.40)In (X, T,U) coordinates, the corresponding nonlinear reaction-diffusion equation(4.1) becomesUT =e?2T UXX + 2XU3X ? 4X ln XU3X4U2X. (4.41)744.3. Examples of inverse potential systems arising from point symmetriesHence, introducing the variables r = UX and s = UT , one obtains the locally re-lated intermediate system of the corresponding nonlinear reaction-diffusion equa-tion (4.1) given byr = UX,s = UT ,s = e?2T rX + 2Xr3 ? 4X ln Xr34r2.(4.42)Excluding U from the intermediate system (4.42), one obtains the inverse potentialsystem of the corresponding nonlinear reaction-diffusion (4.1) given byrT = sX,s = e?2T rX + 2Xr3 ? 4X ln Xr34r2.(4.43)In addition, excluding s from the inverse potential system (4.42), one obtains thelocally related subsystem of the inverse potential system (4.43) given byrT =(e?2T rX + 2Xr3 ? 4X ln Xr34r2)X. (4.44)which is in a CL form. The PDE (4.44) is in a CL form and the nonlinear reaction-diffusion equation (4.1) has no local CL. Thus the PDE (4.44) is nonlocally relatedto the nonlinear reaction-diffusion equation (4.1).The constructed inverse potential systems for the nonlinear reaction-diffusionequation (4.1) (Q(u) = u ln u) are illustrated in Figure 4.4.(4.1)(4.22), (4.23) (4.26)(4.38), (4.39) (4.43), (4.44)Figure 4.4: The constructed inverse potential systems for the nonlinear reaction-diffusion equation (4.1) (Q(u) = u ln u), with the arrows pointing to the inversepotential systems.4.3.2 Nonlinear diffusion equationConsider the scalar nonlinear diffusion equationvt = K (vx) vxx, (4.45)754.3. Examples of inverse potential systems arising from point symmetrieswhere K (vx) is an arbitrary constitutive function. The point symmetry classifica-tion of the locally related PDE system (3.5) of the nonlinear diffusion equation(4.45) is listed in Table 3.2, modulo its group of equivalence transformations. Byprojection of the symmetries in Table 3.2, one obtains that, for arbitrary K (vx),there are are four point symmetries of the nonlinear diffusion equation (4.45),namely, Y1 = ??x , Y2 =??t , Y3 = x??x + 2t??t + v??v and Y4 =??v .(I) Inverse potential system arising from Y1Since the nonlinear diffusion equation (4.45) is invariant under translations of itsindependent variable x, one can interchange x and v to generate an invertibly relatedPDE of the nonlinear diffusion equation (4.45) given byxt =K(1xv)xvvx2v. (4.46)Introducing new variables w = xv and y = xt, one obtains the locally related inter-mediate system of the nonlinear diffusion equation (4.45) given byw = xv,y = xt,y =K(1w)wvw2.(4.47)Excluding x from the PDE system (4.47), one obtains the inverse potential systemof the nonlinear diffusion equation (4.45) given bywt = yv,y =K(1w)wvw2.(4.48)Moreover, one can exclude the variable y from the PDE system (4.48) to obtainthe locally related subsystem of the inverse potential system (4.48) given bywt =????????K(1w)wvw2????????v. (4.49)(II) Inverse potential system arising from Y2Since the nonlinear diffusion equation (4.45) is invariant under translations of itsindependent variable t, one can interchange t and v to obtain an invertibly relatedPDE of the PDE (4.45) given byt2v ? K(? txtv) (2tvtxtxv ? t2xtvv ? t2v txx)= 0. (4.50)764.3. Examples of inverse potential systems arising from point symmetriesIntroducing new variables ? = tv and ? = tx, one obtains the locally related inter-mediate system of the nonlinear diffusion equation (4.45) given by? = tv,? = tx,?2 ? K(? ??) (2???x ? ?2?v ? ?2?x)= 0.(4.51)Excluding t from the PDE system (4.51), one obtains the inverse potential systemof the nonlinear diffusion equation (4.45) given by?x = ?v,?2 ? K(? ??) (2???x ? ?2?v ? ?2?x)= 0.(4.52)(III) Inverse potential system arising from Y3Since the nonlinear diffusion equation (4.45) is invariant under the scaling symme-try generated by Y3 = x ??x + 2t??t + v??v , one can use the corresponding canonicalcoordinate transformation given byX = tx2,T = vx,V = ln x(4.53)to map the nonlinear diffusion equation (4.45) into the invertibly related PDE? VXV2T + K(1 + TVT + 2XVXVT) (?4XVT VT X + VTT + 4XVXVTT ? V2T? 8X2VXVT VT X + 4X2V2XVTT +2XVXV2T + 4X2V2T VXX)= 0.(4.54)Introducing new variables ? = VX and ? = VT , one obtains the locally relatedintermediate system of the nonlinear diffusion equation (4.45) given by? = VX,? = VT ,? ??2 + K(1 + T? + 2X??) (?4X??X + ?T + 4X??T ? ?2?8X2???X + 4X2?2?T + 2X??2 + 4X2?2?X)= 0.(4.55)774.3. Examples of inverse potential systems arising from point symmetriesExcluding V from the intermediate system (4.55), one obtains the inverse potentialsystem of the nonlinear diffusion equation (4.45) given by?T = ?X,? ??2 + K(1 + T? + 2X??) (?4X??X + ?T + 4X??T ? ?2?8X2???X + 4X2?2?T + 2X??2 + 4X2?2?X)= 0.(4.56)(IV) Inverse potential system arising from Y4From its invariance under translations of its dependent variable v, one can applydirectly the symmetry-based method to the equation (4.45). Letting u = vx, z = vt,one obtains the corresponding locally related intermediate system of the nonlineardiffusion equation (4.45) given byu = vx,z = vt,z = K(u)ux.(4.57)Excluding v from the intermediate system (4.57), one obtains the inverse potentialsystem of the nonlinear diffusion equation (4.45) given byut = zx,z = K(u)ux.(4.58)Excluding z from the PDE system (4.58), one obtains the locally related subsystemof the inverse potential system (4.58) given by the nonlinear diffusion equationut = (K (u) ux)x . (4.59)Remark 4.3.1 In fact, the above procedure is in the reverse direction of Exam-ple 3.2.5 and Example 3.2.12, in which the given PDE is the nonlinear diffusionequation (4.59). In particular, in Example 3.2.5 and Example 3.2.12, we start withthe nonlinear diffusion equation (4.59), and use the CL-based method to obtainthe nonlinear diffusion equation (4.45). Conversely, in the above example, thenonlinear diffusion equation (4.45) is the starting PDE. We finally construct thenonlocally related nonlinear diffusion equation (4.59) through the symmetry-basedmethod.784.3. Examples of inverse potential systems arising from point symmetries(IV) Inverse potential system for the the nonlinear diffusion equation (4.59)Now take as the given PDE the nonlinear diffusion equation (4.59). The pointsymmetry classification for the nonlinear diffusion equation (4.59) is presented inTable 3.1, modulo its group of equivalence transformations. There are three pointsymmetries of the nonlinear diffusion equation (4.59) for arbitrary K(u): X1 = ??x ,X2 = ??t and X3 = x??x + 2t??t . Therefore, one can construct three inverse potentialsystems of the nonlinear diffusion equation (4.59) through the symmetry-basedmethod. Take X1 for example. From its invariance under translations in x, one canemploy the hodograph transformation interchanging x and u to obtain the invertiblyrelated PDE of the nonlinear diffusion equation (4.59):xt = ?(K(u)xu)u. (4.60)Accordingly, let p = xu and q = xt, one obtains the following locally relatedintermediate system of the nonlinear diffusion equation (4.59):p = xu,q = xt,q = ?(K(u)p)u.(4.61)Excluding the variable x from the PDE system (4.61), one obtains the inverse po-tential system of the nonlinear diffusion equation (4.59) given bypt = qu,q = ?(K(u)p)u.(4.62)Finally, after excluding the variable q from the PDE system (4.62), one obtains thelocally related subsystem of the inverse potential system (4.62) given bypt = ?(K(u)p)uu. (4.63)The constructed inverse potential system for the nonlinear diffusion equation(4.45) are indicated in Figure 4.5.4.3.3 Nonlinear wave equationAs a third example, we construct a further nonlocally related PDE system of thenonlinear wave equationutt = (c2(u)ux)x, (4.64)794.3. Examples of inverse potential systems arising from point symmetries(4.45)(4.48), (4.49)(4.58), (4.59) (4.56)(4.52)(4.62), (4.63)Figure 4.5: The constructed inverse potential system for the nonlinear diffusionequation (4.45), with the arrows pointing to the inverse potential systems..with an arbitrary constitutive function c(u).In Section 3.2, a tree of equivalent PDE systems was constructed for the nonlin-ear wave equation (4.64). We now use point symmetries of the following potentialsystem of the nonlinear wave equation (4.64):vx = ut,vt = c2(u)ux(4.65)to obtain nonlocally related PDE systems of the nonlinear wave equation (4.64).For arbitrary c(u), the potential system (4.65) has the following point symmetries:Y1 = ??t , Y2 =??x , Y3 =??v , Y4 = x??x + t??t and Y?, where Y? represent theinfinite number of point symmetries arising from the linearization of the potentialsystem (4.65) through the hodograph transformation (interchange of independentand dependent variables).Due to its invariance under translations in v and t, the PDE system (4.65) has apoint symmetry with the infinitesimal generator ??v ???t . Corresponding canonicalcoordinates yield an invertible point transformation of the form:? :???????????????????X = x,T = u,U = t + v,V = v.(4.66)The transformation (4.66) maps the potential system (4.65) into the invertiblyrelated PDE systemVXUT ? VT UX ? 1 = 0,VT + c2(T )UX ? c2(T )VX = 0,(4.67)804.4. Examples of nonlocal symmetries arising from the symmetry-based methodwhich is invariant under translations in U and V .First of all, by excluding the dependent variable V from the PDE system (4.67),one obtains the following subsystem given byUTT + c2(T )(c2(T )UXX ? UXXU2T ? UTT U2X ? 2UT X + 2UT XUT UX)? 2c(T )c?(T )(UX ? U2XUT)= 0.(4.68)which, in turn, is an equivalent PDE for the nonlinear wave equation (4.64).Secondly, by excluding the dependent variable U from the PDE system (4.67),one obtains another subsystem given byc(T )(V2XVTT ? 2VXVT VT X + VXXV2T ? c3(T )VXX)? 2c?(T )V2XVT = 0, (4.69)which, in turn, is another equivalent PDE for the nonlinear wave equation (4.13).Remark 4.3.2 In terms of (x, u, v) coordinates, the equation (4.69) becomesc(u)(v2xvuu ? 2vxvuvux + vxxv2u ? c2(u)vxx)? 2c?(u)v2xvu = 0. (4.70)It is straightforward to check the equation (4.70) is invertibly related to the PDE(3.43) after interchanging x and v. However, in next section we will show that theequation (4.68) is nonlocally related to any PDE system constructed in Example3.2.21.4.4 Examples of nonlocal symmetries arising from thesymmetry-based methodIn the previous section, we constructed several inverse potential systems for thenonlinear reaction-diffusion equation (4.1), the nonlinear diffusion equations (4.45)and (4.59), and the nonlinear wave equation (4.64). For the nonlinear reaction-diffusion equation (4.1), one can show that each point symmetry of the constructedinverse potential systems yields no nonlocal symmetry of the nonlinear reaction-diffusion equation (4.1). In this section, it is shown that for the nonlinear diffusionequations (4.45) and (4.59), and the nonlinear wave equation (4.64), nonlocal sym-metries can arise from some of the constructed inverse potential systems. Mostimportantly, some previously unknown nonlocal symmetries are obtained for thenonlinear wave equation (4.64).814.4. Examples of nonlocal symmetries arising from the symmetry-based method4.4.1 Nonlocal symmetries of nonlinear diffusion equationAs shown in Proposition 3.3.5, the nonlinear diffusion equation (4.59) has a pointsymmetry X6 that induces a nonlocal symmetry of the PDE system (3.5). Sincethe nonlinear diffusion equation (4.45) is locally related to the PDE system (3.5), itfollows that X6 also yields a nonlocal symmetry of the nonlinear diffusion equation(4.45).Now consider the class of scalar PDEs (4.49). The equivalence transformationsfor this class of PDEs arise from the six generatorsE1 =??v, E2 =??w+ 2Kw??K, E3 = w??w+ 2K ??K,E4 = v??v + 2K??K , E5 = t??t ? K??K , E6 =??t .(4.71)Thus the group of equivalence transformations for the class of PDEs (4.49) is givenbyv? = a3v + a1,?t = a5t + a6,w? = a4w + a2,?K =a23(a4w + a2)2a5w2K,(4.72)where a1, . . . , a6 are arbitrary constants with a3a4a5 , 0.In Table 4.2, we present the point symmetry classification of the PDE (4.49),modulo its group of equivalence transformations (4.72).By similar reasoning as in the proof of Proposition 3.3.5, one can show that,for K(u) = u? 23 , the point symmetry V5 of the PDE (4.49) yields a nonlocal sym-metry of the PDE system (4.47), which is locally related to the nonlinear diffusionequation (4.45). Hence V5 yields a nonlocal symmetry of the nonlinear diffusionequation (4.45).Moreover, comparing Tables 3.1 and 4.2, one also sees that when K(u) = u? 23 ,since its infinitesimal component for the variable u has an essential dependenceon the variable v, the symmetry V5 of the PDE (4.49) yields a nonlocal symmetryof the nonlinear diffusion equation (4.59), which cannot be obtained through itspotential system (3.5). By similar reasoning, when K(u) = u?2, one can show thatthe symmetries V6, V7 and V? of the PDE (4.49) yield nonlocal symmetries of thenonlinear diffusion equation (4.59).Remark 4.4.1 Comparing Tables 3.3 and 4.2, for the nonlinear diffusion equation(4.59), one concludes that the nonlocal symmetries yielded by V5, V6, V7 and V?824.4. Examples of nonlocal symmetries arising from the symmetry-based methodTable 4.2: Point symmetry classification for the PDE (4.49)K (1/w) K(u) # admitted point symmetries in (t, v,w) coordinatesarbitrary arbitrary 3 V1 = ??t , V2 =??v , V3 = 2t??t + v??vw?? u? 4 V1, V2, V3, V4 = (2 + ?)v ??v ? 2w ??ww23 u?23 5 V1, V2, V3, V4 (? = ? 23 ), V5 = 3vw ??w ? v2 ??vw2 u?2 ?V1, V2, V3, V4 (? = ?2), V6 = ?vw ??w + 2t ??v ,V7 = 4t2 ??t + 4vt??v ? (2t + v2)w ??w ,V? = G(t, v) ??w , where G(t, v) satisfies Gt = GvvTable 4.2: Point symmetry classification for the PDE (4.49) (continued)K (1/w) K(u) # admitted point symmetries in (t, v, u) coordinatesarbitrary arbitrary 3 V1, V2, V3w?? u? 4 V1, V2, V3, V4 = (2 + ?)v ??v + 2u ??uw23 u?23 5 V1, V2, V3, V4 (? = ? 23 ), V5 = ?3uv ??u ? v2 ??vw2 u?2 ?V1, V2, V3, V4 (? = ?2), V6 = uv ??u + 2t ??v ,V7 = 4t2 ??t + 4vt??v + (2t + v2)u ??u ,V? = ?u2G(t, v) ??u , where G(t, v) satisfiesGt = Gvvcorrespond to the nonlocal symmetries yielded by Z9, Z10, Z11 and Z? respec-tively.In addition, consider the PDE (4.63). The equivalence transformations of theclass of PDEs (4.63) arise from the six generatorsE1 =??u , E2 = u??u + 2K??K , E3 = p??p + 2K??K ,E4 = t??t? K ??K, E5 =??t, E6 = u2??u? 3up ??p? 2Ku ??K.(4.73)Hence, the five-parameter group of equivalence transformations of the PDE class(4.63), arising from the first five generators of (4.73), is given byu? = a2u + a1,?t = a4t + a5,p? = a3 p,?K =a22a23a4K,(4.74)834.4. Examples of nonlocal symmetries arising from the symmetry-based methodwhere a1, . . . , a5 are arbitrary constants with a2a3a4 , 0.The generator E6 yields the additional one-parameter group of equivalencetransformations given byu? = u1 ? a6u,?t = t,p? = (1 ? a6u)3 p,?K = (1 ? a6u)2K,(4.75)where a6 is an arbitrary constant.In Table 4.3, we present the point symmetry classification of the PDE (4.63),modulo its group of equivalence transformations.Table 4.3: Point symmetry classification for the PDE (4.63)K (u) # admitted point symmetriesarbitrary 2 W1 = ??t , W2 = 2t??t + p??pu? 3 W1, W2, W3 = 2u ??u + (? ? 2)p ??peu 3 W1, W2, W4 = 2 ??u + p??p11+u2 e? arctan u 3 W1, W2,W5 = 2(1 + u2) ??u ? p(6u ? ?) ??pu?2 4 W1, W2, W3 (? = ?2), W6 = u2 ??u ? 3pu ??pSimilar to the situation in Proposition 3.3.5, when K(u) = 11+u2 e? arctan u, thepoint symmetry W5 of the PDE (4.63) yields a nonlocal symmetry of the cor-responding intermediate system (4.61), which is locally related to the nonlineardiffusion equation (4.59). Hence W5 yields a nonlocal symmetry of the nonlin-ear diffusion equation (4.59) with K(u) = 11+u2 e? arctan u. By similar reasoning, thesymmetry W6 also yields a nonlocal symmetry of the nonlinear diffusion equation(4.59) with K(u) = u?2.Taking the equivalence transformation (4.75) into consideration, one can ob-tain more nonlocal symmetries for the nonlinear diffusion equation (4.59) fromthe corresponding PDE (4.63). In particular, the equivalence transformation (4.75)maps u? into u??(1 + a6u?)?(?+2), eu into (1 + a6u?)?2eu?1+a6 u? . Moreover, the symme-tries W3 and W4 are mapped into ?W3 and ?W4 respectively. One can show thatwhen K(u) = u?(1 + a6u)?(?+2), ?W3 = 2u(1 + a6u) ??u ? p(6a6u ? ? + 2) ??p ; whenK(u) = (1 + a6u)?2eu1+a6u , ?W4 = 2(1 + a6u)2 ??u ? p(6a26u + 6a6 ? 1) ??p . Similarto the situation in Proposition 3.3.5, one can show that ?W3 and ?W4 yield nonlocalsymmetries of the corresponding nonlinear diffusion equations (4.59).844.4. Examples of nonlocal symmetries arising from the symmetry-based methodRemark 4.4.2 Comparing Tables 3.2 and 4.3, for the nonlinear diffusion equa-tion (4.59), one concludes that when K(u) = 11+u2 e? arctan u, the nonlocal symme-try yielded by W5 corresponds to the nonlocal symmetry yielded by Y9. WhenK(u) = u?2, the nonlocal symmetry yielded by W6 corresponds to a nonlocal sym-metry yielded by Y?.4.4.2 Nonlocal symmetries of nonlinear wave equationWe now use the PDE (4.68) to find previously unknown nonlocal symmetries ofthe nonlinear wave equation (4.64).In [3], the point symmetry classification is given for the class of nonlinear waveequations (4.64), which is presented in Table 4.4, modulo its group of equivalencetransformations:x? = a1x + a4,?t = a2t + a5,u? = a3u + a6,c? = a1a2c,(4.76)where a1, . . . , a6 are arbitrary constants with a1a2a3 , 0.Table 4.4: Point symmetry classification for the nonlinear wave equation (4.64)c(u) # admitted point symmetriesarbitrary 3 X1 = ??x , X2 =??t , X3 = x??x + t??tu? (? , 0) 4 X1, X2, X3, X4 = ?x ??x + u ??ueu 4 X1, X2, X3, X5 = x ??x +??uu?2 5 X1, X2, X3, X4 (? = ?2), X6 = t2 ??t + tu ??uu?23 5 X1, X2, X3, X4 (? = ? 23 ), X7 = x2 ??x ? 3xu ??uThe equivalence transformations for the PDE class (4.68) arise from the fivegeneratorsE1 =??T , E2 =??X , E3 =??U ,E4 = T??T + X??X + U??U , E5 = ?T??T + X??X + c??c .(4.77)Correspondingly, the five-parameter group of equivalence transformations for the854.4. Examples of nonlocal symmetries arising from the symmetry-based methodclass of PDEs (4.68) is given by?T = a4a5T + a1,?X = a4a5X + a2,?U = a4U + a3,c? = a5c,(4.78)where a1, . . . , a5 are arbitrary constants with a4a5 , 0.The point symmetry classification of the PDE (4.68), modulo its equivalencetransformations (4.78), is presented in Table 4.5.Table 4.5: Point symmetry classification for the PDE (4.68)c(T ) c(u) # admitted point symmetries in (X, T,U) coordinatesarbitrary arbitrary 3 W1 =??U , W2 =??X ,W3 = (X +? Tc2(?)d?) ??X + U ??UT? u? 4 W1, W2, W3 (c(T ) = T?),W4 = T ??T + (2? + 1)X ??X + (? + 1)U ??UeT eu 4 W1, W2, W3 (c(T ) = eT ),W5 = ??T + 2X??X + U??UT?2 u?2 5 W1, W2, W3 (c(T ) = T?2), W4 (? = ?2),W6 = U2 ??U + TU??T ?UT 3??XT?23 u?23 5 W1, W2, W3 (c(T ) = T? 23 ), W4 (? = ? 23 ),W7 = (XT ? 3T 23 ) ??T + (XT?13 ? X23 ) ??XTable 4.5: Point symmetry classification for the PDE (4.68) (continued)c(T ) c(u) # (x, u) components of admitted symmetriesarbitrary arbitrary 3 ?W2 = ??x , ?W3 = (x +? uc2(?)d?) ??xT? u? 4?W2, ?W3 (c(u) = u?),?W4 = u ??u + (2? + 1)x ??xeT eu 4?W2, ?W3 (c(u) = eu),?W5 = ??u + 2x??xT?2 u?2 5?W1, ?W2, ?W3 (c(u) = u?2), ?W4 (? = ?2),?W6 = u(t + v) ??u ? t+vu3 ??xT?23 u?23 5?W1, ?W2, ?W3 (c(u) = u? 23 ), ?W4 (? = ? 23 ),?W7 = (xu ? 3u 23 ) ??u + (xu?13 ? x23 ) ??x864.4. Examples of nonlocal symmetries arising from the symmetry-based methodRemark 4.4.3 In order to determine whether a symmetry W of the PDE (4.68)yields a nonlocal symmetry of the nonlinear wave equation (4.64), it requires usto trace back to the nonlinear wave equation (4.64) using the PDE system (4.67).Because one excludes the dependent variable V from the potential system (4.67),one needs to investigate how the variable V changes under the action induced by W.Since ??1? ( ??V ) = ??v ? ??t , where ??1 is the inverse of the transformation (4.66), theinfinitesimal components for the variables x and u remain invariant when tracingback. This is why we only present the (x, u) components of admitted symmetriesin Table 4.5 (continued).Proposition 4.4.4 The symmetries W6 and W7 yield nonlocal symmetries of thePDE system (4.65).Proof. If the symmetry W6 yields a local symmetry ?W6 of the potential system(4.65) with c(u) = u?2, then, in evolutionary form, ?W6 =(U2 ? TUUT + UT 3 UX)??U+F[U,V] ??V , where the differential function F[U,V] must depend on X, T , U, Vand the partial derivatives of U and V with respect to X and T . By applying ?W6to the corresponding PDE system (4.67) which is invertibly related to the potentialsystem (4.65), one can show that F[U,V] must be of the form F(X, T,U,V,UX,UT ).Applying ?W(?)6 to the corresponding PDE system (4.67) and making appropriatesubstitutions, one can prove that the resulting determining equation system is in-consistent. Hence W6 yields a nonlocal symmetry of the potential system (4.65)with c(u) = u?2.By similar reasoning, it turns out that W7 also yields a nonlocal symmetry ofthe potential system (4.65) with c(u) = u? 23 . When c(u) is arbitrary, in (x, t, u, v) coordinates, W3 = (x+? uc2(?)d?) ??x + (t+v) ??t . One can show that W3 is a point symmetry of the potential system (4.65),whose infinitesimal component for the variable t has an essential dependence onv. By projection, W3 yields a nonlocal symmetry of the nonlinear wave equation(4.64).When c(u) = u?2, the infinitesimal components for the variables (x, u) of thesymmetry W6 depend on the variable v. By Remark 4.4.3, W6 yields a nonlocalsymmetry of the nonlinear wave equation (4.64).When c(u) = u? 23 , if the symmetry W7 yielded a local symmetry ?W7 of thenonlinear wave equation (4.64), then ?W7 = ?W7 + f [u] ??t , where the function f [u]depends on x, t, u and its derivatives. Since ??1? ( ??V ) = ??v ? ??t , when tracing back tothe PDE system (4.65), the infinitesimal component for the variable v must be equalto ? f [u]. Thus W7 would also yield a local symmetry of the PDE system (4.65),which is a contradiction since W7 yields a nonlocal symmetry of the PDE system874.5. Summary(4.65). Hence, W7 yields a nonlocal symmetry of the nonlinear wave equation(4.64).Remark 4.4.5 One can show that the symmetries W4 and W5 yield point sym-metries ?W4 = W4 + (? + 1)V ??V and ?W5 = W5 + V ??V of the PDE system (4.67)respectively since in terms of (x, t, u, v) coordinates, ?W4 = u ??u + (2?+1)x ??x + (?+1)t ??t + (? + 1)v ??v = (2? + 1)Y4 ? Y5 and ?W5 = ??u + 2x ??x + t ??t + v ??v = Y4 + Y7.Hence, by projection, W4 and W5 yield point symmetries of the nonlinear waveequation (4.64).Remark 4.4.6 Comparing the symmetries listed in [24], one sees that the symme-tries W6 and W7 yield previously unknown nonlocal symmetries of the nonlinearwave equation (4.64).4.5 SummaryIn this chapter, we introduced a new systematic symmetry-based method for con-structing nonlocally related PDE systems (inverse potential systems) of a givenPDE system. The starting point for this method is any point symmetry of the givenPDE system. In the case of three or more independent variables, the symmetry-based method directly yields determined nonlocally related systems for a givenPDE system, unlike the situation in the CL-based method where one must appendgauge constraints.The symmetry-based method was shown to yield previously unknown nonlo-cally related systems for nonlinear reaction-diffusion, nonlinear diffusion and non-linear wave equations. In addition, it was shown that nonlocally related symmetry-based systems could yield nonlocal symmetries of a given PDE system. Moreover,some previously unknown nonlocal symmetries were obtained for the nonlinearwave equation.88Chapter 5New Exact Nonclassical Solutionsof the NLK Equation5.1 IntroductionAn exact solution is of great interest for researchers, since it plays an essentialrole in the analysis of a PDE system. A significant application of symmetries of agiven PDE system is the finding of exact solutions of the PDE system. The methodof using a local symmetry to construct exact solutions of a PDE system is calledLie?s classical method [21, 25, 29, 39, 53, 75, 80]. Moreover, reduction througha point symmetry could lead to the solution of a boundary value problem for agiven PDE system [16, 21]. In [15, 27], Lie?s classical method was generalized tothe nonclassical method, in which one searches for ?nonclassical symmetries? of agiven PDE system. In particular, ?nonclassical symmetries? are local symmetriesof an augmented PDE system consisting of the given PDE system, the invariantsurface condition and their differential consequences. It follows that ?nonclassicalsymmetries? leave only submanifolds of solutions invariant. Consequently, thenonclassical method turns out to be useful for finding further specific solutions inaddition to those obtained by Lie?s classical method.In this chapter, we first present the basic ideas of both Lie?s classical methodand the nonclassical method. Then we apply the nonclassical method to obtainpreviously unknown exact solutions of the dimensional NLK equation [60] givenbyut = x?2(x4(?ux + ?u + ?u2))x, (5.1)where ? > 0, ? ? 0 and ? > 0 are arbitrary constants.The NLK equation (5.1), also known as the photon diffusion equation, wasfirst presented by Kompaneets [60], and in dimensionless form, after appropriatescalings of x, t and u, can be written asut = x?2(x4(ux + u + u2))x, (5.2)andut = x?2(x4(ux + u2))x, (5.3)895.2. Lie?s classical methodwhen ? , 0 and the case with dominating induced scattering ? = 0 (u2 ? u),respectively. By construction, the solutions obtained by the nonclassical method ofcourse include the solutions obtained by Ibragimov [55] through classical symme-try reductions. Correspondingly, these new solutions yield five families of solutionswith initial conditions of physical interest. It is shown that three of these familiesof solutions exhibit quiescent behaviour, i.e., limt??u(x, t) = 0, and that the othertwo families of solutions exhibit blow up behaviour, i.e., limt?t?u(x, t?) = ? for somefinite t? depending on a constant in their initial conditions. Moreover, we considernontrivial stationary solutions of (5.3). We exhibit four families of stationary so-lutions not presented explicitly in [50] for the NLK equations (5.2) and (5.3). Weshow that two of these families of stationary solutions are unstable.5.2 Lie?s classical method5.2.1 The invariant form methodIn this section, we present the invariant form method for constructing invariantsolutions of a given PDE system [21, 25, 29, 39, 53, 75]. Consider a PDE systemR{x; u} of order l with n independent variables x = (x1, . . . , xn) and m dependentvariables u = (u1, . . . , um), given byR?[u] = R?(x, u, ?u, . . . , ?lu) = 0, ? = 1, . . . , s. (5.4)Suppose the PDE system R{x; u} (5.4) has a point symmetry with the infinitesimalgeneratorX =n?i=1?i(x, u) ??xi+m?j=1? j(x, u) ??u j. (5.5)Definition 5.2.1 A solution u = f (x), with u? = f ?(x), ? = 1, . . . , m, of thePDE system R{x; u} (5.4) is an invariant solution arising from the point symmetry(5.5) if u? = f ?(x) is an invariant surface of the point symmetry (5.5) for each? = 1, . . . ,m.From Definition 5.2.1, a function u = f (x) is an invariant solution of the PDEsystem R{x; u} (5.4) arising from the point symmetry (5.5) if and only if u = f (x)satisfies the following two conditions:X(u? ? f ?(x))???u= f (x) = 0, ? = 1, . . . ,m. (5.6)R?(x, u, ?u, . . . , ?lu)???u= f (x) = 0, ? = 1, . . . , s. (5.7)905.2. Lie?s classical methodIn order to obtain the invariants of the point symmetry (5.5) of the PDE systemR{x; u} (5.4), one can employ the characteristic method stated in Chapter 2, i.e.,solve the following characteristic equations:dx1?1(x, u) = ? ? ? =dxn?n(x, u) =du1?1(x, u) = ? ? ? =dum?m(x, u) . (5.8)Suppose one obtains n + m ? 1 corresponding functionally independent invariantsgiven by?1(x, u), . . . , ?n?1(x, u), ?1(x, u), . . . , ?m(x, u), (5.9)with?(?1, . . . , ?n?1)?(xi1 , . . . , xin?1 ) , 0,?(?1, . . . , ?m)?(u1, . . . , um) , 0. (5.10)By introducing the new independent variables y = (y1, . . . , yn) and dependent vari-ables v = (v1, . . . , vm):yi = ?i(x, u), i = 1, . . . , n ? 1,yn = ?n(x, u),v? = ??(x, u), ? = 1, . . . ,m,(5.11)withX?n(x, u) = 1,one obtains the canonical coordinates corresponding to the point symmetry (5.5).The point transformation corresponding to the canonical coordinates (5.14) mapsthe point symmetry (5.5) into the canonical form?X = ??yn . (5.12)Suppose the PDE system R{x; u} (5.4) becomes the transformed PDE system S{y; v}in the canonical coordinates (5.14). Since the transformed PDE system S{y; v}has the translation point symmetry with the infinitesimal generator ?X, it followsthat the transformed PDE system S{y; v} does not depend explicitly on the newindependent variable yn. Consequently, the transformed PDE system S{y; v} hasparticular solutions of the formv? = h?(y1, . . . , yn?1), ? = 1, . . . ,m. (5.13)By the assumption (5.10), one can solve xi1 , . . . , xin?1 and u1, . . . , um from(5.14) in terms of y = (y1, . . . , yn?1), v = (v1, . . . , vm) and the remaining variablexin , i.e.,xik = ?ik (xin , y, v), k = 1, . . . , n ? 1,u? = ??(x, v) = ???(xin , y, v), ? = 1, . . . ,m. (5.14)915.2. Lie?s classical methodTherefore, for each solution v = h(y), where h(y) = (h1(y), . . . , hm(y)), of the trans-formed PDE system S{y; v}, there is a corresponding implicit solutionu? = ???(xin , ?(x, u), h(?(x, u))), ? = 1, . . . ,m, (5.15)of the given PDE system R{x; u} (5.4), where ?(x, u) = (?i1 (x, u), . . . , ?in?1(x, u)).Moreover, one can show that the solution (5.15) is invariant under the point sym-metry (5.5). In particular, if the point symmetry is of the formX =n?i=1?i(x) ??xi+m?j=1? j(x, u) ??u j, (5.16)then one can choose y = ?(x). It follows that the solution (5.15) becomesu? = ???(xin , ?(x), h(?(x))), ? = 1, . . . ,m, (5.17)which defines an explicit invariant solution of the given PDE system R{x; u} (5.4).From the above discussion, one concludes that the invariant solutions of a givenPDE system can be found by solving a reduced DE system involving fewer inde-pendent variables.Example 5.2.2 Consider the heat equationut = uxx, (5.18)which has the point symmetryX = 2t ??x? xu ??u. (5.19)The invariants of X are given by? = t, ? = uex24t , (5.20)for t > 0. By introducing the canonical coordinates corresponding to X:y1 = t,y2 =x2t,v = uex24t ,(5.21)one obtains the invertibly related PDE of (5.18) given by4y21vy1 + 2y1v ? vy2y2 = 0. (5.22)925.2. Lie?s classical methodIn order to obtain the invariant solution corresponding to X, one seeks solutions ofPDE (5.22) of the form v = h(y1). Consequently, the PDE (5.22) reduces to theODE2y21hy1 + y1h = 0. (5.23)The solution of the reduced ODE (5.23) is given byh(y1) = c?y1 , (5.24)where c is an arbitrary constant. Hence the invariant solution corresponding to Xis given byu = h(y1)e?x24t = c?te?x24t . (5.25)One can replace the point symmetry (5.5) by any local symmetry with the in-finitesimal generator?X =m?j=1Q j[u] ??u j(5.26)to obtain invariant solutions arising from a local symmetry (5.26). One can refer to[29] for more details of the extension of Lie?s classical method for finding invariantsolutions to local symmetries.The invariant form method can be applied to a boundary value problem for aPDE system provided the symmetry of the given PDE system also leaves invariantthe boundary and the boundary conditions [16, 21, 29]. Moreover, this methodcan also be applied to the nonlocally related PDE systems of a given PDE system,which possibly yields further exact solutions of the given PDE system [41].5.2.2 The direct substitution methodIf one is unable to solve the characteristic equations (5.8), one can employ the directsubstitution method to resolve such a dilemma [25]. Without loss of generality, onecan assume that ?n(x, u) , 0. From the condition (5.6), it follows that??(x, u) ?n?i=1?i(x, u)?u??xi= 0, ? = 1, . . . ,m. (5.27)Solving for ?u??xn from the system of equations (5.27), one obtains?u??xn =??(x, u)?n(x, u) ?n?1?i=1?i(x, u)?n(x, u)?u??xi= 0, ? = 1, . . . ,m. (5.28)935.3. The nonclassical methodThus the terms involving derivatives of u with respect to xn appearing in the PDEsystem R{x; u} (5.4) can be expressed in terms of x, u and derivatives of u withrespect to x? = (x1, . . . , xn?1). Then one obtains a reduced DE system involvingn? 1 independent variables x?, m dependent variables u, and the parameter variablexn. A solution u = f (x?; xn) of the reduced DE system yields an invariant solutionof the PDE system R{x; u} (5.4) provided the equations (5.28) are satisfied.5.3 The nonclassical methodIn the nonclassical method, for a given PDE system one seeks symmetries thatleave only a submanifold of the solution manifold invariant. Such a ?nonclassicalsymmetry? maps solution surfaces not in the submanifold to surfaces that are notsolutions of the PDE system.Consider a PDE system R{x; u} (5.4). In the nonclassical method, instead ofseeking local symmetries of the PDE system R{x; u} (5.4), one seeks local symme-tries that leave invariant a submanifold of the solution manifold of the PDE systemR{x; u} (5.4). In particular, one seeks functions ?i(x, u), ? j(x, u), i = 1, . . . , n, j = 1,. . . , m, so thatX =n?i=1?i(x, u) ??xi+m?j=1? j(x, u) ??u j(5.29)is a ?symmetry? (?nonclassical symmetry?) of the submanifold, which is the aug-mented PDE system A{x; u} consisting of the given PDE system R{x; u} (5.4), theinvariant surface condition equationsI?(x, u, ?u) ? ??(x, u) ?n?i=1?i(x, u)?u??xi= 0, ? = 1, . . . ,m, (5.30)and their differential consequences. Consequently, one obtains an overdeterminedset of nonlinear determining equations for the unknown functions ?i(x, u), ? j(x, u),i = 1, . . . , n, j = 1,. . . , m.For any given ?nonclassical symmetry?, one can employ either the invariantform or the direct substitution method to find the invariant solution correspondingto such a ?nonclassical symmetry?.Definition 5.3.1 A solution u = f (x) of a given PDE system R{x; u} (5.4) is anonclassical solution if u = f (x) arises from a ?nonclassical symmetry? of R{x; u}(5.4), and does not arise as an invariant solution of the given PDE system R{x; u}(5.4) with respect to its local symmetries.945.4. Nonclassical analysis of the NLK equationIndeed, for any functions ?i(x, u), ? j(x, u), i = 1, . . . , n, j = 1, . . . ,m, the fol-lowing expressionsX(1)I?(x, u, ?u) =m?j=1(????u j?n?i=1??i?u j?u??xi) ? I j, ? = 1, . . . ,m, (5.31)where X(1) is the first order prolongation of X, vanish on I?(x, u, ?u) = 0, ? =1, . . . ,m. Therefore, the nonclassical method includes Lie?s classical method.In the nonclassical method, invariance of a given PDE system R{x; u} (5.4) isreplaced by invariance of the augmented PDE system A{x; u}. Consequently, it ispossible to find symmetries leaving invariant the augmented PDE system A{x; u}which are not symmetries of a given PDE system R{x; u} (5.4). In turn, this canlead to further exact solutions of a given PDE system R{x; u} (5.4).When a given PDE is a scalar PDE with two independent variables, one needsonly to consider two essential cases when solving the determining equations for(?(x, t, u), ?(x, t, u), ?(x, t, u)). Let x1 = x, x2 = t, ?1 = ?(x, t, u), ?2 = ?(x, t, u).If the infinitesimal generator X = ?(x, t, u) ??x + ?(x, t, u) ??t + ?(x, t, u) ??u generatesa ?nonclassical symmetry? of the PDE system R{x; u} (5.4), then so does Y =p(x, t, u)X, where p(x, t, u) is any smooth function. It follows that if ? , 0, one canset ? ? 1, so that only two cases need to be considered: ? ? 1 and ? ? 0, ? ? 1.In [45], Clarkson and Kruskal introduced a method (the direct method) to ob-tain exact solutions of a scalar PDE with two independent variables. In the directmethod, one aims to find exact solutions of the formu(x, t) = ?(x, t,w(z)) with z = z(x, t), (5.32)where ? and z are functions of the indicated variables. By substituting (5.32) intothe given PDE, one obtains an ODE. After solving the resulting ODE, one canobtain exact solutions for the given PDE. In [74], it was shown that the nonclas-sical method is more general than the direct method. Further discussions of thenonclassical method can be found in [46, 47, 51, 65, 77, 78].Other discussions and extensions of obtaining exact solutions of a given PDEsystem appear in [22, 41, 83, 89]5.4 Nonclassical analysis of the NLK equationIn this section, we use the nonclassical method to obtain new exact solutions of theNLK equation [36].955.4. Nonclassical analysis of the NLK equation5.4.1 Invariant solutions of the NLK equationAs stated in Ibragimov [55], the NLK equation (5.1) describes an interaction offree electrons and electromagnetic radiation, specifically, the interaction of a low-energy homogeneous photon gas with a rarefied electron gas through Comptonscattering. In equation (5.1), u is the density of the photon gas (photon numberdensity), t is a dimensionless time and x = h?kT , where h is Planck?s constant and? is the photon frequency. Then h? denotes the photon energy. T is the elec-tron temperature and k is Boltzmann?s constant. The terms u and u2 in equation(5.1) correspond to spontaneous scattering (Compton effect) and induced scatter-ing, respectively [71, 95]. The Kompaneets model has been investigated in manypublications, and some numerical and analytical solutions have been obtained forthe NLK equation (5.1) [13, 50, 55, 71, 72, 86, 88, 95, 96].By applying Lie?s classical method to the NLK equation (5.3), one is able toobtain its corresponding invariant solutions. In [55], it was shown that the NLKequation (5.3) has two point symmetriesX1 =??t, X2 = x??x? u ??u.Using these two point symmetries, Ibragimov obtained two sets of invariant solu-tions given byu(x, t) = 1x(1 ? a1e2t), (5.33)where a1 is an arbitrary constant, andu(x, t) = ?(z)xwith z = xea2t, (5.34)where a2 is an arbitrary constant and ?(z) satisfies the ODEz??? + (2? + 2 ? a2)?? + 2z(?2 ? ?) = 0.5.4.2 Nonclassical symmetries of the NLK equationThe nonclassical method is now applied to the NLK equations (5.2) and (5.3) re-spectively. Here the invariant surface condition equation becomes?(x, t, u)ux + ?(x, t, u)ut = ?(x, t, u). (5.35)(I) Nonclassical symmetries of the NLK equation (5.2)(I-a) The case when ? ? 1965.4. Nonclassical analysis of the NLK equationThe nonclassical method applied to (5.2) yields the following determining equationsystem for the infinitesimals (?(x, t, u), ?(x, t, u)).2?x? ? 8xu? ?2??x? 4x? + 4u2? + 4u? + ?t ? x2?xx ? 4x?x + 4xu2?u? 2x2u?x ? x2?x ? 8xu2?x + 4xu?u ? 8xu?x = 0,4? ? 2x2? + 2?2x? ?t ? 2x2u?x ? 2x2?xu + 2?u? ? 12xu2?u ? 12xu?u? 4x?x + x2?xx ? x2?x ? 2??x = 0,2x2?xu ? 2x2?u ? 4x2u?u ? 8x?u ? x2?uu ? 2?u? = 0,x2?uu = 0.(5.36)The solution of the determining equation system (5.36) is given by????????(x, t, u) = 0,?(x, t, u) = 0. (5.37)Hence the corresponding ?nonclassical symmetry? isY1 =??t,which directly results from the point symmetry X1.(I-b) The case when ? ? 0 and ? ? 1In this case, the determining equation for ?(x, t, u) is given by4xu?u ? x2?2?uu ? 2x2??xu ? 4u ? 4u2 ? 4? + ?t ? x2?xx ? 6x?x ? 6x?? 2x2?2 ? x2?x + 4xu2?u ? 2x2u?x ? 12xu? ? 2x??u = 0.(5.38)One is unable to find the general solution of (5.38). Hence one must consideransa?tze to obtain particular solutions of (5.38). If one considers an ansatz of theform ? = f (x, t) + g(x, t)u + h(x, t)u2, one obtains?(x, t, u) = b1x4? u ? u2, (5.39)where b1 is an arbitrary constant. The corresponding ?nonclassical symmetry? isY2 =??x+ (b1x4? u ? u2) ??u.975.4. Nonclassical analysis of the NLK equation(II) Nonclassical symmetries of the NLK equation (5.3)(II-a) The case when ? ? 1The nonclassical method applied to (5.3) yields the following determining equationsystem for the infinitesimals (?(x, t, u), ?(x, t, u)).4u2? ? 8xu2?x ?2??x? 2x2u?x ? 8xu? ? x2?xx + ?t+ 4xu2?u + 2?x? ? 4x?x = 0,x2?xx ? 12xu2?u ? 2x2u?x ? 2?x? ? ?t ? 2x2? + 4?+ 2?2x+ 2?u? ? 2x2?xu ? 4x?x = 0,2x2?xu ? 4x2u?u ? 8x?u ? 2??u ? x2?uu = 0,x2?uu = 0.(5.40)The solutions of the determining equation system (5.40) are given by????????(x, t, u) = b2x,?(x, t, u) = ?b2u,(5.41)where b2 is an arbitrary constant, and????????(x, t, u) = ?2x2u,?(x, t, u) = 4xu2 ? 2u. (5.42)The solution (5.41) yields the ?nonclassical symmetry?Y3 = b2x??x +??t ? b2u??u ,which directly results from the point symmetry X1 + b2X2. The solution (5.42)yields the ?nonclassical symmetry?Y4 = ?2x2u??x+ ??t+ (4xu2 ? 2u) ??u,which does not result from any point symmetry of (5.3).(II-b) The case when ? ? 0 and ? ? 1In this case, the determining equation for ?(x, t, u) is given by? 4? ? 4u2 ? 6x?x ? 2x??u ? 12xu? ? 2x2?2 ? 2x2u?x + ?t+ 4xu2?u ? x2?xx ? 2x2??xu ? x2?2?uu = 0.(5.43)985.4. Nonclassical analysis of the NLK equationIf one considers an ansatz of the form ? = f (x, t) + g(x, t)u + h(x, t)u2, the deter-mining equation (5.43) has solutions?(x, t, u) = b4e?2tx2(b3 + b4e?2t ? x)+ (x ? 2b3 ? 2b4e?2t)ux(b3 + b4e?2t ? x), (5.44)?(x, t, u) = 1x2(1 + b5e2t)? 2ux, (5.45)?(x, t, u) = b6x4? u2, (5.46)where b3, b4, b5 and b6 are arbitrary constants.Hence, the corresponding ?nonclassical symmetries? are given byY5 =??x +[b4e?2tx2(b3 + b4e?2t ? x)+ (x ? 2b3 ? 2b4e?2t)ux(b3 + b4e?2t ? x)]??u ,Y6 =??x+[1x2(1 + b5e2t)? 2ux]??u,andY7 =??x + (b6x4? u2) ??u .5.4.3 New exact solutions of the NLK equationIt is obvious that the invariant solutions arising from Y1 and Y3 are those obtainedby Ibragimov [55], given by solutions (5.33) and (5.34). Moreover, the invariantsolution corresponding to Y2 is the stationary solution obtained by Dubinov [50]for the NLK equation (5.2).Consider the ?nonclassical symmetry? Y4 of the NLK equation (5.3). Usingthe direct substitution method, one seeks solutions of the PDE system{ut = 4x(ux + u2) + x2(uxx + 2uux), (5.47)ut = 2x2uux + (4xu2 ? 2u). (5.48)After equating the right hand sides of (5.47) and (5.48), one obtains4xux + x2uxx + 2u = 0. (5.49)The solution of (5.49) is given byu(x, t) = A(t) + B(t)xx2, (5.50)995.4. Nonclassical analysis of the NLK equationwhere A(t) and B(t) are arbitrary functions. Substituting (5.50) into (5.47), oneobtains an ordinary differential equation (ODE) system for A(t) and B(t):{A?(t) + 2A(t) ? 2A(t)B(t) = 0, (5.51)B?(t) + 2B(t) ? 2B(t)2 = 0. (5.52)From (5.52), one obtains B(t) ? 0 or B(t) = 11+c1e2t , where c1 is an arbitrary con-stant. In particular, there are three families of solutions when B(t) . 0. In terms ofan arbitrary constant t0, ?? < t0 < ?, these solutions are given byB(t) = 12[1 ? tanh(t + t0)] , where 0 < B(t) < 1;B(t) = 12[1 ? coth(t + t0)] , where???????B(t) < 0, if t > ?t0,B(t) > 1, if t < ?t0;B(t) ? 1.If B(t) . 0, one has A(t) = ?cB(t), where c is an arbitrary constant. If B(t) ? 0,one has A(t) = c2e?2t, where c2 is an arbitrary constant. Therefore, there are fourfamilies of solutions of (5.3).F1 : u(x, t) = x ? c2x2 [1 ? tanh(t + t0)] ; (5.53)F2 : u(x, t) = x ? c2x2 [1 ? coth(t + t0)] ; (5.54)F3 : u(x, t) = x ? cx2; (5.55)F4 : u(x, t) = c2x2e2t. (5.56)The first two families of solutions F1 and F2 are new and cannot be obtainedthrough classical symmetry reductions.The corresponding initial conditions u(x, 0) = U(x) are given by the following.(I) The family F1:(I-a) U(x) = b(x?c)x2with 0 < b < 1, c ? 0, on the domain 0 < x < ?. Such a U(x)is illustrated in Figure 5.1 (a). The corresponding solutions of (5.3) are given byu(x, t) = x ? c2x2[1 ? tanh(t + t0)] (5.57)1005.4. Nonclassical analysis of the NLK equationwith constants t0 = 12 ln(1b ? 1), 0 < b < 1 and c ? 0, valid for x > 0, t > 0. Foreach value of x, the solution u(x, t) is monotonically decreasing as a function oft. Moreover limt??u(x, t) = 0 for any x > 0. The evolution of a solution u(x, t) isillustrated in Figure 5.1 (b).(a) (b)Figure 5.1: (a) U(x) = b(x?c)x2, 0 < b < 1, c ? 0, x > 0. In (b), u(x, t) is given by(5.57) for x > 0, t > 0, with the arrow pointing in the direction of increasing t.(I-b) U(x) = b(x?c)x2with 0 < b < 1, c > 0, on the domain x ? c. Such a U(x) isillustrated in Figure 5.2 (a). The corresponding solutions of (5.3) are given byu(x, t) = x ? c2x2[1 ? tanh(t + t0)] (5.58)with constants t0 = 12 ln(1b ? 1), 0 < b < 1 and c > 0, valid for x ? c, t > 0. Foreach value of x, the solution u(x, t) is monotonically decreasing as a function oft. Moreover limt??u(x, t) = 0 for any x ? c. The evolution of a solution u(x, t) isillustrated in Figure 5.2 (b).(II) The family F2 :(II-a) U(x) = b(x?c)x2with b < 0, c > 0, on the domain 0 < x ? c. Such a U(x) isillustrated in Figure 5.3 (a). The corresponding solutions of (5.3) are given byu(x, t) = x ? c2x2[1 ? coth(t + t0)] (5.59)1015.4. Nonclassical analysis of the NLK equation(a) (b)Figure 5.2: (a) U(x) = b(x?c)x2, 0 < b < 1, c > 0, x ? c. In (b), u(x, t) is given by(5.58) for x ? c, t > 0, with the arrow pointing in the direction of increasing t.with constants t0 = 12 ln(1 ? 1b), b < 0 and c > 0, valid for 0 < x ? c, t > 0. Foreach value of x, the solution u(x, t) is monotonically decreasing as a function oft. Moreover limt??u(x, t) = 0 for 0 < x ? c. The evolution of a solution u(x, t) isillustrated in Figure 5.3 (b).(II-b) U(x) = b(x?c)x2with b > 1, c > 0, on the domain x ? c. Such a U(x) isillustrated in Figure 5.4 (a). The corresponding solutions of (5.3) are given byu(x, t) = x ? c2x2[1 ? coth(t + t0)] (5.60)with constants t0 = 12 ln(1 ? 1b), b > 1 and c > 0, valid for x ? c, 0 < t < ?t0. Foreach value of x, the solution u(x, t) is monotonically increasing as a function of t.Moreover limt??t0u(x, t) = ? for each value of x ? c. The evolution of a solutionu(x, t) is illustrated in Figure 5.4 (b).(II-c) U(x) = b(x?c)x2with b > 1, c ? 0, on the domain x > 0. Such a U(x) isillustrated in Figure 5.5 (a). The corresponding solutions of (5.3) are given byu(x, t) = x ? c2x2[1 ? coth(t + t0)] (5.61)with constants t0 = 12 ln(1 ? 1b), b > 1 and c ? 0, valid for x > 0, 0 < t < ?t0. Foreach value of x, the solution u(x, t) is monotonically increasing as a function of t.1025.4. Nonclassical analysis of the NLK equationMoreover limt??t0u(x, t) = ? for each value of x > 0. The evolution of a solutionu(x, t) is illustrated in Figure 5.5 (b).(a) (b)Figure 5.3: (a) U(x) = b(x?c)x2, b < 0, c > 0, 0 < x ? c. In (b), u(x, t) is given by(5.59) for 0 < x ? c, t > 0, with the arrow pointing in the direction of increasing t.(a) (b)Figure 5.4: (a) U(x) = b(x?c)x2, b > 1, c > 0, x ? c. In (b), u(x, t) is given by (5.60)for 0 < x ? c, 0 < t < ?t0, with the arrow pointing in the direction of increasing t.1035.4. Nonclassical analysis of the NLK equation(a) (b)Figure 5.5: (a) U(x) = b(x?c)x2, b > 1, c ? 0, x > 0. In (b), u(x, t) is given by (5.61)for 0 < x ? c, 0 < t < ?t0, with the arrow pointing in the direction of increasing t.5.4.4 Stationary solutionsStationary solutions of the NLK equation (5.1) were found in [50] in terms ofthe doubly degenerate Heun?s function (HeunD) and its derivative (HeunD?). Astationary solution u(x, t) ? V(x) of the NLK equation (5.3) satisfies the ODEV ?(x) + V2 = c3x4(5.62)for some constant c3 which represent the photon flux in the frequency domain. Onecan show that a nontrivial stationary solutionV(x) = b(x ? c)x2(5.63)satisfies equation (5.62) for some constant Q if and only if b = 1 and c is anarbitrary constant. Consequently, c3 = c2. Interestingly, the explicit family ofsolutions V(x) = x?cx2is not exhibited in [50]. For c > 0, V(x) is exhibited in Figure5.6 (a); for c ? 0, V(x) is exhibited in Figure 5.6 (b).From the solutions obtained in the last subsection, one sees that all of thesenontrivial stationary solutions are unstable since a slight change in the initial con-dition will lead to a solution blowing up in finite time or decaying to the trivialstationary solution u(x, t) ? 0 as t ? ?.1045.4. Nonclassical analysis of the NLK equation(a) (b)Figure 5.6: The stationary solution V(x) = x?cx2, in (a) c > 0, in (b) c ? 0.Moreover, if one applies the ?nonclassical symmetry? Y7 to the NLK equation(5.3), one obtains two more families of explicit stationary solutions, F5 and F6.The family of stationary solutions F5, in terms of an arbitrary positive constanta, is given byF5 : V(x) =x + a tan(ax)x2, (5.64)valid on the domains:(1) x > 2api , illustrated in Figure 5.7 (a);(2) 2a(2k+1)pi < x ? xk, where xk ?(2a(2k+1)pi ,2a(2k?1)pi)satisfies xk + a tan(axk)= 0,k = 1, 2, . . . , illustrated in Figure 5.7 (b).The family of stationary solutions F6, in terms of an arbitrary positive constanta, is given byF6 : V(x) =x ? a tanh(ax)x2, (5.65)valid on the domain x ? ?, where ? is the unique positive solution of the equation? ? a tanh(a?)= 0. The maximum value of V(x) occurs at x = ? = 2a1+LambertW(e?1) ,in terms of the Lambert W function. Such a solution is illustrated in Figure 5.8.1055.5. Summary(a) (b)Figure 5.7: The stationary solution V(x) = x+a tan(ax )x2, in (a) x > 2api , in (b) 2a(2k+1)pi <x ? xk.Figure 5.8: The stationary solution x?a tanh(ax )x2, x ? ?.5.5 SummaryIn this chapter, we presented Lie?s classical method and the nonclassical method forthe construction of exact solutions of a PDE system. Then we used the nonclassicalmethod to obtain some previously unknown solutions of the NLK equation (5.3)1065.5. Summary[36]. These solutions do not arise as invariant solutions of the NLK equation (5.3)with respect to its point symmetries. Moreover, the newly obtained exact solutionsare explicit solutions of (5.3) expressed in terms of elementary functions. It wasfurther observed that these solutions explicit both quiescent and blow up behaviourdepending on their initial conditions. It was also shown that related stationarysolutions are unstable.107Chapter 6Concluding Chapter6.1 ConclusionsIn this thesis, we presented the basic ideas of symmetries, CLs and their appli-cations. In particular, we focused on nonlocally related PDE systems and theirapplications, and the application of the nonclassical method. The following newresults were obtained.(1) In Theorem 3.2.7, for two potential systems S1{x, t; u, v} and S2{x, t; u,w},arising from two nontrivial and linearly independent local CLs of a givenPDE system R{x, t; u}, we showed that if S1{x, t; u, v} and S2{x, t; u,w} arein Cauchy-Kovalevskaya form, then the potential variable v is a nonlocalvariable of S2{x, t; u,w} and the potential variable w is a nonlocal variable ofS1{x, t; u, v}.(2) In Section 3.5, we investigated the relationship between local symmetriesof a given PDE system and those of its potential systems. In particular, inTheorem 3.5.1, we proved that any local symmetry of a given PDE systemhaving precisely n local CLs is a projection of some local symmetry of itsn-plet potential system.(3) In Chapter 4, we introduced a new systematic symmetry-based method toconstruct nonlocally related PDE systems (inverse potential systems) fora given PDE system. The symmetry-based method is complementary tothe well known CL-based method. Most importantly, the symmetry-basedmethod can be directly applied to any PDE system that has a point symme-try, no matter whether it has nontrivial local CLs, and no matter how manyindependent variables it involves. In addition, by applying the symmetry-based method, we constructed previously unknown nonlocally related PDEsystems (inverse potential systems) for the nonlinear reaction-diffusion equa-tions, the nonlinear diffusion equations and the nonlinear wave equations.Moreover, we also showed that for these example equations, one can obtainnonlocal symmetries (including previously unknown nonlocal symmetries)from some of their inverse potential systems.1086.2. Future work(4) In Chapter 5, we applied the nonclassical method to the NLK equation. Con-sequently, we obtained four families of solutions, two of which are new andcannot be obtained by the classical symmetry reductions. Moreover, we an-alyze the behaviour of the solutions with initial condition U(x) = b(x?c)x2. Ithas been shown that a slight change of c in the initial condition will lead tosignificant change of the solutions of the NLK equation. In particular, forthree cases of U(x) = b(x?c)x2, the solutions of the NLK equation exhibit qui-escent behaviour, and for two cases of U(x) = b(x?c)x2, the solutions of theNLK equation exhibit blow up behaviour. Finally, we obtained some newstationary solutions of the NLK equation.6.2 Future workBesides the results presented in the thesis, there are still some open problems thatarise from the work presented in this thesis.6.2.1 To determine whether two PDE systems are nonlocally relatedIn Chapters 3 and 4, we presented two different systematic methods for the con-struction of nonlocally related PDE systems of a given PDE system: the CL-basedmethod and the symmetry-based method. In [23], an extended procedure for theconstruction of a tree of nonlocally related PDE systems was introduced. However,as stated in Remark 3.2.17, it may be difficult to determine whether two resultingsystems are nonlocally related. The first step to solve this problem would be to in-vestigate the existence of an invertible transformation that relates two PDE systemin a resulting tree. There are several possible methods to deal with the problem.(1) To investigate the number of point symmetries (contact symmetries) of suchPDE systems (scalar PDEs). If two PDE systems (scalar PDEs) are relatedby an invertible transformation, then they must have the same number ofpoint symmetries (contact symmetries). It follows that if two PDE systems(scalar PDEs) in a resulting tree have a different number of point symme-tries (contact symmetries), then they cannot be mapped to each other by aninvertible transformation.(2) To investigate the number of multipliers or local CLs of such PDE systems.If two PDE systems are related by an invertible transformation, then theymust have the same number of multipliers or local CLs. It follows that iftwo PDE systems in a resulting tree have a different number of multipliers1096.2. Future workor local CLs, then they cannot be mapped to each other by an invertibletransformation.(3) Cartan?s method of equivalence (see [58, 76] and references therein for moredetails).In general, to investigate whether two PDE systems in a tree are nonlocallyrelated, one needs to investigate whether there exists a local function that relatesthe two PDE systems. Is the existence of a local function related to the existenceof an invertible transformation? Can we find necessary and sufficient conditions todetermine whether two PDE systems are nonlocally related?6.2.2 The existence of subsystemsNonlocally related PDE systems can arise from subsystems of a given PDE sys-tem. However, not all PDE systems can generate a subsystem through excludingsome dependent variables. Can we find necessary and sufficient conditions so thata given PDE system generates subsystems by excluding dependent variables? Thisproblem is also related to the extended procedure for constructing nonlocally re-lated PDE systems. To obtain more nonlocally related PDE systems, one can em-ploy invertible transformations acting on a given PDE system. Which invertibletransformations can lead to PDE systems that have subsystems? Is there a relation-ship between the existence of a nonlocally related subsystem and the existence ofa symmetry of the PDE system yielding the subsystem?6.2.3 The relationship of symmetries of a given PDE system andthose of its potential systemsIn [19], a conjecture about the construction of potential systems and the the rela-tionship of symmetries of a given PDE system and those of its potential systemswas presented. In the case of two independent variables, the conjecture is as fol-lows.Conjecture 6.2.1 (1) The process of obtaining potential systems S(1) = S(1){x, t;u, v(1)}, S(2) = S(2){x, t; u, v(1), v(2)}, . . . , S(N) = S(N){x, t; u, v(1), v(2), . . . , v(N)}of a given PDE system R{x, t; u} terminates at some finite N where either? S(N) can be linearized by some invertible point transformation; or? S(N) has no further conservation law.(2) The group of all point symmetries of S(N) yields, through projections, allpoint symmetries of any subsystem of S(N) including R{x, t; u}, S(1), . . . ,S(N?1).1106.2. Future workFor the first conjecture, a key step would be to develop a counterpart of Theo-rem 2.3.13, i.e., to determine whether one can find all local CLs of a PDE system.For the second conjecture, we presented a related result in Section 3.5. In partic-ular, we proved that if a given PDE system R{x, t; u} has precisely n local CLs,then the local symmetries of its n-plet potential system yield all local symmetriesof R{x, t; u}. A related question is whether one can find a relationship between lo-cal symmetries of each k-plet (1 ? k ? n) potential system of R{x, t; u} and thoseof the n-plet potential system. More specifically, can each local symmetry of anyk-plet potential system be obtained by projection of some local symmetry of then-plet potential system? If the answer to this question is no, does there exist a po-tential system whose local symmetries includes all local symmetries of each k-pletpotential system?6.2.4 The application of the obtained nonlocal symmetriesIn Chapters 3 and 4, we obtained some nonlocal symmetries of each given PDE.Can these nonlocal symmetries yield new exact solutions of each correspondinggiven PDE? Can we use such nonlocal symmetries to obtain more useful inversepotential systems? It is meaningful to continue to investigate these problems in thefuture.6.2.5 Nonlocal symmetries for PDE systems with three or moreindependent variablesThe symmetry-based method for constructing nonlocally related PDE systems canbe directly applied to PDE systems with three or more independent variables. Forthe CL-based method, in order to obtain nonlocal symmetries of a given PDEsystem with three or more independent variables from potential systems arisingfrom divergence-type CLs, it is necessary to add gauge constraints to such under-determined potential systems. However, there is no known systematic procedureto determine which gauge constraints yield nonlocal symmetries. Since the inversepotential systems constructed by the symmetry-based method are determined, itis expected that for some PDE systems with three or more independent variables,one should be able to directly obtain nonlocal symmetries from local symmetriesof corresponding inverse potential systems. 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