Stability and instability of standing waves for the generalized Davey-Stewartson system

Abstract

We study the stability and instability properties of standing waves for the equation $iu_t+\Delta u+a\vert u\vert^{p-1}u+E_1(\vert u\vert^2)u=0$ in $\Bbb R^2$ or $\Bbb R^3$, which derives from the generalized Davey-Stewartson system in the elliptic-elliptic case. We show that if $n=2$ and $a(p-3)<0$, then the standing waves generated by the set of minimizers for the associated variational problem are stable. We also show that if $n=3$, $a>0$ and $1+4/3<p<5$ or $a<0$ and $1<p<3$, then the standing waves are strongly unstable. We employ the concentration compactness principle due to Lions and the compactness lemma due to Lieb to solve the associated minimization problem.