On Attraction to Solitons in Relativistic Nonlinear Wave Equations

We discuss a new general conjecture on attractors and soliton-type asymptotics of the solutions to nonlinear wave and Klein-Gordon Eqns, with a general Lie group $G$, in an infinite space. For the case of relativistic invariant equations, the conjecture reads as follows: every finite energy solution decays to a finite number of solitons combined with a dispersive wave. The asymptotics are inspired by the Bohr Quantum Transitions and de Broglie's Wave-Particle Duality.

We explain the physical motivations and suggestions, list known results and describe our numerical experiments.