Sharp thresholds for stability and instability of standing waves in a double power nonlinear Schr\"odinger equation

TL;DR

This paper determines the precise conditions under which standing waves in a double power nonlinear Schrödinger equation are stable or unstable, especially focusing on the critical thresholds related to the frequency parameter.

Contribution

It provides the explicit formula for the limit of the derivative of the squared L2 norm of standing waves as frequency approaches zero, establishing sharp stability and instability thresholds.

Findings

Explicit formula for limit_{\u03c9 o 0} \u2202_ \u2225  _^2

Sharp thresholds for stability and instability of standing waves

Stability properties change at critical frequency values

Abstract

We study the stability/instability of standing waves for the one dimensional nonlinear Schr\"odinger equation with double power nonlinearities: \begin{align*} &i\partial_t u +\partial_x^2 u -|u|^{p-1}u +|u|^{q-1}u=0, \quad (t,x)\in \mathbb{R}\times\mathbb{R} ,~1<p<q. \end{align*} When $q<5$, the stability properties of standing waves $e^{i\omega t}\phi_{\omega}$ may change for the frequency $\omega$. A sufficient condition for yielding instability for small frequencies are obtained in previous results, but it has not been known what the sharp condition is. In this paper we completely calculate the explicit formula of $\lim_{\omega\to0}\partial_{\omega}\|\phi_{\omega}\|_{L^2}^2$, which is independent of interest, and establish the sharp thresholds for stability and instability of standing waves.