In this article we consider nonlinear Schrödinger (NLS) equations in for , , and . We consider nonlinearities satisfying a flatness condition at zero and such that solitary waves are stable. Let be solitary wave solutions of the equation with different speeds . Provided that the relative speeds of the solitary waves are large enough and that no interaction of two solitary waves takes place for positive time, we prove that the sum of the is stable for in some suitable sense in . 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 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 solitons of the gKdV equations by the authors of the present article (see [15, Th. 1(i)]).
"Stability in of the sum of solitary waves for some nonlinear Schrödinger equations." Duke Math. J. 133 (3) 405 - 466, 15 June 2006. https://doi.org/10.1215/S0012-7094-06-13331-8