Stability in H1 of the sum of K solitary waves for some nonlinear Schrödinger equations

Abstract

In this article we consider nonlinear Schrödinger (NLS) equations in ${\mathbb{R}}^d$ for $d=1$, $2$, and $3$. We consider nonlinearities satisfying a flatness condition at zero and such that solitary waves are stable. Let $R_k(t,x)$ be $K$ solitary wave solutions of the equation with different speeds $v_1,v_2,\dots ,v_K$. Provided that the relative speeds of the solitary waves $v_k-v_{k-1}$ are large enough and that no interaction of two solitary waves takes place for positive time, we prove that the sum of the $R_k(t)$ is stable for $t\ge 0$ in some suitable sense in $H^1$. To prove this result, we use an energy method and a new monotonicity property on quantities related to momentum for solutions of the nonlinear Schrödinger equation. This property is similar to the $L^2$ monotonicity property that has been proved by Martel and Merle for the generalized Korteweg–de Vries (gKdV) equations (see [12, Lem. 16, proof of Prop. 6]) and that was used to prove the stability of the sum of $K$ solitons of the gKdV equations by the authors of the present article (see [15, Th. 1(i)]).