Instability of the standing waves for the nonlinear Klein-Gordon equations in one dimension

TL;DR

This paper investigates the stability of standing wave solutions to the one-dimensional nonlinear Klein-Gordon equation, proving that these solutions are orbitally unstable at the critical frequency, extending previous high-dimensional results.

Contribution

It establishes the orbital instability of standing waves at the critical frequency in one dimension, filling a gap left by prior high-dimensional analyses.

Findings

Standing waves are orbitally unstable at the critical frequency in 1D.

Extends instability results from higher dimensions to one dimension.

Provides mathematical proof using virial identities and Sobolev inequalities.

Abstract

In this paper, we consider the following nonlinear Klein-Gordon equation \begin{align*} \partial_{tt}u-\Delta u+u=|u|^{p-1}u,\qquad t\in \mathbb{R},\ x\in \mathbb{R}^d, \end{align*} with $1<p< 1+\frac{4}{d}$. The equation has the standing wave solutions $u_\omega=e^{i\omega t}\phi_{\omega}$ with the frequency $\omega\in(-1,1)$, where $\phi_{\omega}$ obeys \begin{align*} -\Delta \phi+(1-\omega^2)\phi-\phi^p=0. \end{align*} It was proved by Shatah (1983), and Shatah, Strauss (1985) that there exists a critical frequency $\omega_c\in (0,1)$ such that the standing waves solution $u_\omega$ is orbitally stable when $\omega_c<|\omega|<1$, and orbitally unstable when $|\omega|<\omega_c$. Further, the critical case $|\omega|=\omega_c$ in the high dimension $d\ge 2$ was considered by Ohta, Todorova (2007), who proved that it is strongly unstable, by using the virial identities and the radial Sobolev inequality. The one dimension problem was left after then. In this paper, we consider the one-dimension problem and prove that it is orbitally unstable when $|\omega|=\omega_c$.