Geodesic
Asymptotic stability of solitons for nonlinear hyperbolic equations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 68 (2013) no. 2, pp. 283-334

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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. Bibliography: 45 titles.
Keywords: nonlinear hyperbolic equations, asymptotic stability, relativistic invariance, Hamiltonian structure, symplectic projection, invariant manifold, kink, Fermi's golden rule, scattering of solitons, asymptotic state.
Mots-clés : soliton 
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E. A. Kopylova. Asymptotic stability of solitons for nonlinear hyperbolic equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 68 (2013) no. 2, pp. 283-334. http://geodesic.mathdoc.fr/item/RM_2013_68_2_a2/