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Existence and stability of fronts in inhomogeneous wave equations

Banff International Research Station for Mathematical Innovation and Discovery

Models describing waves in anisotropic media or media with imperfections usually have inhomogeneous terms. Examples of such models can be found in many applications, for example in nonlinear optical waveguides, water waves moving over a bottom with topology, currents in nonuniform Josephson junctions, DNA-RNAP interactions etc. Travelling waves in such models tend to interact with the inhomogeneity and get trapped, reflected, or slowed down. In this talk, wave equations with finite length inhomogeneities will be considered, assuming that the spatial domain can be written as the union of disjoint intervals, such that on each interval the wave equation is homogeneous. The underlying Hamiltonian structure allows for a rich family of stationary front solutions and the values of the energy (Hamiltonian) in each intermediate interval provide natural parameters for the family of orbits. It will be shown that changes of stability can only occur at critical points of the length of the inhomogeneity as a function of the energy density inside the inhomogeneity and we give a necessary and sufficient criterion for the change of stability. These results will be illustrated with some examples.

Mathematics; Partial differential equations; Dynamical systems and ergodic theory; Functional analysis

video/mp4

Author affiliation: University of Surrey

2017-12-20

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/64110

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/