Asymptotic stability of solitons for nonlinear hyperbolic equations

Fundamental results on asymptotic stability of solitons are surveyed, methods for proving asymptotic stability are illustrated based on the example of a nonlinear relativistic wave equation with Ginzburg–Landau potential. Asymptotic stability means that a solution of the equation with initial data close to one of the solitons can be asymptotically represented for large values of the time as a sum of a (possibly different) soliton and a dispersive wave solving the corresponding linear equation. The proof techniques depend on the spectral properties of the linearized equation and may be regarded as a modern extension of the Lyapunov stability theory. Examples of nonlinear equations with prescribed spectral properties of the linearized dynamics are constructed.
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