On Global attraction to solitary waves for Klein-Gordon equation with concentrated nonlinearity

TL;DR

This paper proves that solutions to a 3D Klein-Gordon equation with concentrated nonlinearity tend to solitary waves over time, revealing the underlying spectral mechanism driving this global attraction.

Contribution

It establishes the convergence of finite energy solutions to solitary waves for the Klein-Gordon equation with point nonlinearity, introducing a spectral analysis approach.

Findings

Solutions asymptotically approach solitary waves

Spectral gap analysis shows spectrum reduces to a single frequency

Mechanism involves nonlinear energy transfer and dispersion

Abstract

The global attraction is proved for the nonlinear 3D Klein-Gordon equation with a nonlinearity concentrated at one point. Our main result is the convergence of each "finite energy solution" to the manifold of all solitary waves as $t\to\pm\infty$. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersion radiation. We justify this mechanism by the following strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time-spectrum in the spectral gap $[-m,m]$ and satisfies the original equation. Then the application of the Titchmarsh Convolution Theorem reduces the spectrum of each omega-limit trajectory to a single frequency $\omega\in[-m,m]$.