On Attraction to Solitons in Relativistic Nonlinear Wave Equations

We describe our numerical experiments on soliton-type asymptotics of solutions to relativistic nonlinear wave equations. We propose a unifying conjecture on minimal global attractors and soliton-type asymptotics of the solutions to nonlinear G-invariant wave and Klein-Gordon Eqns, with a general Lie group G, in an infinite space. For the case of relativistic equations, invariant with respect to the Lorentz group, the conjecture reads as follows: every finite energy solution decays to the sum of a finite combination of solitons and a dispersive wave. The asymptotics are inspired by the Bohr Quantum Transitions and de Broglie’s Wave-Particle Duality. We discuss the physical motivations and suggestions, list known results corre-sponding to four simplest Lie groups, and describe our numerical observations for the Lorentz-invariant equations. Mathematical proof of the observed asymptotics is still open problem