Global attraction to solitary waves in nonlinear dispersive systems

Abstract:
We consider the U(1)-invariant Klein-Gordon equation in one dimension, coupled to finitely many anharmonic oscillators. This is an example of a nonlinear Hamiltonian system with dispersion. Such systems are of considerable interest, in particular in Quantum Field Theory. One expects that in a rather general situation the long time asymptotics for any finite energy solution could be described as a superposition of solitary waves plus dispersion.

We prove that in our model the weak global attractor consists of solitary waves only if the distances between the oscillators are sufficiently small. In the case of larger distances, the global attractor has a more complicated structure due to the presence of trapped modes. We present an example of such an attractor.

Physically, the global attraction to the set of solitary waves is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.

The main analytical tools are the Paley-Wiener arguments and the Titchmarsh Convolution Theorem.

This is a joint work with Alexander Komech (University of Vienna, Austria).