On global attraction to solitary waves for the Klein-Gordon field coupled to several nonlinear oscillators

Abstract

The global attraction is established for all finite energy solutions to a model $\mathbf{U}(1)$-invariant nonlinear Klein-Gordon equation in one dimension coupled to a finite number of nonlinear oscillators: We prove that {\it each finite energy solution} converges as $t\to\pm\infty$ to the set of all ``nonlinear eigenfunctions'' of the form $ϕ(x)e\sp{-iωt}$ if all oscillators are strictly nonlinear, and the distances between all neighboring oscillators are sufficiently small. The {\it global attraction} is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation. This result for one oscillator was obtained in [KK07]. We construct counterexamples showing that the convergence to the solitary waves may break down if the distance between some of the neighboring oscillators is sufficiently large or if some of the oscillators are harmonic. In these cases, the global attractor can contain ``multifrequency solitary waves'' or linear combinations of distinct solitary waves.