Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension

Vianney Combet

doi:10.3934/dcds.2014.34.1961

Article Contents

2014, Volume 34, Issue 5: 1961-1993. Doi: 10.3934/dcds.2014.34.1961

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Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension

Vianney Combet^1,

1.
Université Lille 1, U.F.R. de Mathématiques, 59 655 Villeneuve d'Ascq Cédex,

Received: September 2010

Revised: July 2013

Published: May 2014

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Abstract

For the $L^2$ supercritical generalized Korteweg-de Vries equation, we proved in [2] the existence and uniqueness of an $N$-parameter family of $N$-solitons. Recall that, for any $N$ given solitons, we call $N$-soliton a solution of the equation which behaves as the sum of these $N$ solitons asymptotically as $t \to +\infty$. In the present paper, we also construct an $N$-parameter family of $N$-solitons for the supercritical nonlinear Schrödinger equation in dimension $1$. Nevertheless, we do not obtain any classification result; but recall that, even in subcritical and critical cases, no general uniqueness result has been proved yet.

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Mathematics Subject Classification: Primary: 35Q55, 35Q51; Secondary: 35B40, 37K40.

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