Lecture note 24/2005
Lectures on Global Attractors of Hamilton Nonlinear Wave Equations
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Submission date: 10. Apr. 2005 (revised version: May 2005)
MSC-Numbers: 37K, 35B41, 35L, 35Q, 35P25, 81, 78
PACS-Numbers: 02., 02.30.Jr, 03., 11.10.-z, 11.30.-j, 11.30.Cp, 30., 40.
Keywords and phrases: global attractor, stationary state, soliton, hyperbolic, nonlinear, symmetry group, scattering, couled maxwell-schoedinger equations, couled maxwell-dirac equations, relativistic klein-gordon equation, adiabatic asymptotics
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This is a the survey on recent results on global attractors of the Hamilton nonlinear wave equations. We propose a unifying conjecture concerning the global attractors and long-time asymptotics of finite energy solutions of a class of nonlinear wave equations (Klein-Gordon, Maxwell, Schroedinger, Dirac, coupled systems, etc) with a Lie symmetry group, in an infinite space:
For a generic equation with a Lie symmetry group, each finite energy
solution converges, in the long-time limit, to the sum of finite
combination of solitary waves and a dispesive wave.
The conjecture was proved in the last decade for a list of equations: for 1D nonlinear wave and Klein-Gordon equations, for nonlinear systems of 3D wave, Klein-Gordon and Maxwell equations coupled to a classical particle, Maxwell-Landau-Lifschitz-Gilbert Equations. The equations correspond to the following four symmetry groups:
i) trivial group , ii) translation group ,
iii) rotation group U(1), and iv) rotation group SO(3).
We formulate our results which justify the conjecture for model equations with the four symmetry groups, and describe our numerical experiments for the Lorentz-invariant equations. We explain main ideas of proofs and discuss open problems. Main role in the proof plays an analysis of energy radiation to infinity.
The investigation is inspired by mathematical problems of Quantum Mechanics: Bohr's Transitions to Quantum Stationary States and de-Broglie's Wave-Particle Duality. The conjecture is still open problem for coupled Maxwell-Dirac and Maxwell-Schrödinger Equations. The equations correspond to the translation, rotation and the Lorentz symmetry groups.