Sharp thresholds for stability and instability of standing waves in a double power nonlinear Schrödinger equation

Abstract

We study the stability/instability of standing waves for the one dimensional nonlinear Schrödinger 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 $e^{i\omega t}{\varphi }_{\omega }$ 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 ${\lim }_{\omega \to 0}{\partial }_{\omega }\Vert {\varphi }_{\omega }{\Vert }_{L^2}^2$, which is independent of interest, and establish the sharp thresholds for stability and instability of standing waves.

11 pages. A similar result was obtained independently in arXiv:2112.06529 which appeared on 13 Dec 2021