Duke Mathematical Journal

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

Yvan Martel, Frank Merle, and Tai-Peng Tsai

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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 vkvk1 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 t0 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)]).

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Duke Math. J., Volume 133, Number 3 (2006), 405-466.

First available in Project Euclid: 13 June 2006

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Zentralblatt MATH identifier

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 35Q51: Soliton-like equations [See also 37K40] 35B35: Stability


Martel, Yvan; Merle, Frank; Tsai, Tai-Peng. Stability in $H^1$ of the sum of $K$ solitary waves for some nonlinear Schrödinger equations. Duke Math. J. 133 (2006), no. 3, 405--466. doi:10.1215/S0012-7094-06-13331-8. https://projecteuclid.org/euclid.dmj/1150201198

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