Asymptotic stability of solitons for nonlinear hyperbolic equations

作者：Kopylova E A^*

来源：Russian Mathematical Surveys, 2013, 68(2): 283-334.

DOI：10.1070/RM2013v068n02ABEH004830

• nonlinear hyperbolic equations
• asymptotic stability
• relativistic invariance
• Hamiltonian structure
• symplectic projection
• invariant manifold
• soliton
• kink
• Fermi%26apos
• s golden rule
• scattering of solitons
• asymptotic state

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.

• 出版日期2013