Attractors of Hamiltonian nonlinear wave equations

We consider long time asymptotics and attractors of all finite energy solutions to Hamiltonian nonlinear wave equations in the whole space. We prove long-time convergence to stationary states for the solutions in a Fréchet topology defined by local energy seminorms. This means that the set of stationary states is a point attractor for the systems in this topology.

The convergence in the global norm is impossible due to energy conservation. For translation-invariant systems the convergence implies the soliton-like asymptotics in some global norms.

The convergence is established for one-dimensional nonlinear wave equations with nonlinear terms concentrated at a single point /1/, at a finite set of points, /2/, and at a finite segment, for three-dimensional scalar wave equation coupled to a particle, /3/ and for Maxwell-Lorentz system with a charge, /4/. We establish the soliton-like asymptotics for finite energy solutions to the translation-invariant three-dimensional scalar wave equation coupled to a particle, /5/.

Liapunov-type criterion is established for asymptotic stability in the Fréchet topology of the stationary states of general nD nonlinear wave equations and systems, /6/.

The investigation is inspired by an outstanding role of the stationary states in many phenomena described by nonlinear Hamiltonian wave equations: N. Bohr's postulate on transitions to stationary states in quantum systems, E. Schrödinger's wave mechanics, L. de Broglie's wave-particle duality and radiative damping in classical electrodynamics, /7/.

References

1. A. Komech, On stabilization of string-nonlinear oscillator interaction, J. Math. Anal. Appl., v. 196 (1995), 384-409.
2. A. Komech, On the stabilization of string-oscillators interaction, Russian Journal of Mathematical Physics, 3 (1995), 227-248.
3. A. Komech, H. Spohn and M. Kunze, Long-time asymptotics for a classical particle interacting with a scalar wave field, Comm. Partial Diff. Equs. 22 (1997), 307-335.
4. A. Komech, H. Spohn, Long-time asymptotics for the coupled Maxwell-Lorentz equations, submitted to Comm. Partial Diff. Equs.
5. A. Komech, H. Spohn, Soliton-like asymptotics for a classical particle interacting with a scalar wave field, Nonlinear Analysis 33 (1998), 13-24.
6. A. Komech, B. R. Vainberg, On asymptotic stability of stationary solutions to nonlinear wave and Klein-Gordon equations, Arch. Rat. Mech. Anal. 134 (1996), 227-248.
7. A. Komech, On transitions to stationary states in Hamiltonian nonlinear wave equations, Phys. Letters A 241 (1998), 311-323.