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

In this article we consider nonlinear Schrödinger (NLS) equations in Rd for d=1, 2, and 3. We consider nonlinearities satisfying a flatness condition at zero and such that solitary waves are stable. Let Rk(t,x) be K solitary wave solutions of the equation with different speeds v1,v2,…,vK. Provided that the relative speeds of the solitary waves vk−vk−1 are large enough and that no interaction of two solitary waves takes place for positive time, we prove that the sum of the Rk(t) is stable for t≥0 in some suitable sense in H1. 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 L2 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)]).