Weak attractor of the Klein-Gordon field in discrete space-time interacting with a nonlinear oscillator

TL;DR

This paper proves that solutions of a discretized nonlinear Klein-Gordon equation in space-time tend to a set of multifrequency solitary waves over time, revealing the nonlinear energy transfer mechanisms involved.

Contribution

It introduces a new proof of convergence to solitary waves for a discretized Klein-Gordon equation using quasimeasures and a novel Titchmarsh convolution theorem adaptation.

Findings

Solutions converge to multifrequency solitary waves as time approaches infinity.

The solitary wave manifold components are generically two- or four-dimensional.

Energy transfer from lower harmonics to the continuous spectrum causes attraction.

Abstract

We consider the U(1)-invariant nonlinear Klein-Gordon equation in discrete space and discrete time, which is the discretization of the nonlinear continuous Klein-Gordon equation. To obtain this equation, we use the energy-conserving finite-difference scheme of Strauss-Vazquez. We prove that each finite energy solution converges as $t\to\pm\infty$ to the finite-dimensional set of all multifrequency solitary wave solutions with one, two, and four frequencies. The components of the solitary manifold corresponding to the solitary waves of the first two types are generically two-dimensional, while the component corresponding to the last type is generically four-dimensional. The attraction to the set of solitary waves is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent radiation. For the proof, we develop the well-posedness for the nonlinear wave equation in discrete space-time, apply the technique of quasimeasures, and also obtain the version of the Titchmarsh convolution theorem for distributions on the circle.