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A group analysis of nonlinear differential equations

University of British Columbia

A necessary and sufficient condition is established for the existence of an invertible mapping of a system of nonlinear differential equations to a system of linear differential equations based on a group analysis of differential equations. It is shown how to construct the mapping, when it exists, from the invariance group of the nonlinear system. It is demonstrated that the hodograph transformation, the Legendre transformation and Lie's transformation of the Monge-Ampere equation are obtained from this theorem. The equation (ux)Puxx-uyy=0 is studied and it is determined for what values of p this equation is transformable to a linear equation by an invertible mapping. Many of the known non-invertible mappings of nonlinear equations to linear equations are shown to be related to invariance groups of equations associated with the given nonlinear equations. A number of such; examples are given, including Burgers' equation uxx +uuz-ut=0 a nonlinear diffusion equation (u⁻²ux ) x -ut =0, equations of wave propagation {Vy-wx=0, Vy-avw-bv-cw=0}, equations of a fluid flow {wy+vx=0, wx -v⁻¹wP=0} and the Liouville equation uxy=eu. As another application of group analysis, it is shown how conservation laws associated with the Korteweg-deVries equation, the cubic Schrodinger equation, the sine-Gordon equation and Hamilton's field equation are related to the invariance groups of the respective equations. All relevant background information is in the thesis, including an appendix on the known algorithm for computing the invariance group of a given system of differential equations.

Text

2010-03-30

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http://hdl.handle.net/2429/23079

Mathematics

University of British Columbia

Graduate