Strong Instability of Standing Waves for the Nonlinear Schrödinger Equation in Trapped Dipolar Quantum Gases

Abstract

In this paper, we study the strong instability of standing waves for the nonlinear Schrodinger equation arising in trapped dipolar quantum gases. Two cases are considered: the first when the system is free, the second when a partial/complete harmonic potential is added. In the free case, we present a new argument to prove that the ground state standing waves are strongly unstable by blow-up. In the second case, if $$\partial ^2_\mu S_\omega (Q^{\mu }_\omega )|_{\mu =1}\le 0$$ , we deduce that the ground state standing wave $$u(t,x)=e^{i\omega t}Q_\omega (x)$$ is strongly unstable by blow-up, where $$S_\omega $$ is the action, and $$Q_\omega ^{\mu }=\mu ^{3/2}Q_\omega (\mu x)$$ is the $$L^2$$ -invariant scaling.