On instability of standing waves for the mass-supercritical fractional nonlinear Schr\"odinger equation

TL;DR

This paper proves that ground state standing waves for the focusing mass-supercritical fractional nonlinear Schrödinger equation are strongly unstable and blow up, extending the understanding of stability in different regimes.

Contribution

It establishes the strong instability of ground state standing waves in the mass-supercritical regime using localized virial estimates, complementing previous stability results.

Findings

Ground states are strongly unstable by blow-up.

Instability proven for specific parameter ranges.

Extends stability analysis to supercritical fractional NLS.

Abstract

We consider the focusing $L^2$-supercritical fractional nonlinear Schr\"odinger equation \[ i\partial_t u - (-\Delta)^s u = -|u|^\alpha u, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, \] where $d\geq 2, \frac{d}{2d-1} \leq s <1$ and $\frac{4s}{d}<\alpha<\frac{4s}{d-2s}$. By means of the localized virial estimate, we prove that the ground state standing wave is strongly unstable by blow-up. This result is a complement to a recent result of Peng-Shi [J. Math. Phys. 59 (2018), 011508] where the stability and instability of standing waves were studied in the $L^2$-subcritical and $L^2$-critical cases.