Nonexistence of small, odd breathers for a class of nonlinear wave equations

TL;DR

This paper proves that small, odd breather solutions do not exist for a broad class of nonlinear wave equations with odd nonlinearities, including certain Klein-Gordon models, by showing such solutions decay to zero.

Contribution

It establishes the nonexistence of small, odd breather solutions for a wide class of nonlinear wave equations, addressing a specific open question in the field.

Findings

Small, odd solutions decay to zero in the energy norm.

Nonexistence of small, odd breathers for classical Klein-Gordon equations.

Partially answers an open question about breathers in cubic NLKG.

Abstract

In this note, we show that for a large class of nonlinear wave equations with odd nonlinearities, any globally defined odd solution which is small in the energy space decays to $0$ in the local energy norm. In particular, this result shows nonexistence of small, odd breathers for some classical nonlinear Klein Gordon equations such as the sine Gordon equation and $\phi^4$ and $\phi^6$ models. It also partially answers a question of Soffer and Weinstein in \cite[p. 19]{MR1681113} about nonexistence of breathers for the cubic NLKG in dimension one.