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Spatial decay of multi-solitons of the generalized Korteweg-de Vries and nonlinear Schrödinger equations

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Raphaël Côte
Raphaël Côte
Raphaël Côte
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Xavier Friederich
Xavier Friederich
Xavier Friederich
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Abstract

We study pointwise spatial decay of multi-solitons of the generalized Korteweg-de Vries equations. We obtain that, uniformly in time, these solutions and their derivatives decay exponentially in space on the left of and in the solitons region, and prove rapid decay on the right of the solitons. We also prove the corresponding result for multi-solitons of the nonlinear Schrödinger equations, that is, exponential decay in the solitons region and rapid decay outside.
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Mathematische Annalen (2023) 387:1163–1198
https://doi.org/10.1007/s00208-022-02484-8
Mathematische Annalen
Spatial decay of multi-solitons of the generalized
Korteweg-de Vries and nonlinear Schrödinger equations
Raphaël Côte1·Xavier Friederich1
Received: 31 May 2022 / Revised: 15 September 2022 / Accepted: 18 September 2022 /
Published online: 1 October 2022
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022
Abstract
We study pointwise spatial decay of multi-solitons of the generalized Korteweg-de
Vries equations. We obtain that, uniformly in time, these solutions and their derivatives
decay exponentially in space on the left of and in the solitons region, and prove
rapid decay on the right of the solitons. We also prove the corresponding result for
multi-solitons of the nonlinear Schrödinger equations, that is, exponential decay in
the solitons region and rapid decay outside.
Mathematics Subject Classification Primary 35B40 ·35Q53 ·35Q55; Secondary
35B65 ·37K40
1 Introduction
1.1 (gKdV) multi-solitons
We consider the generalized Korteweg-de Vries equations
∂tu+∂x(∂2
xu+up)=0 (gKdV)
where (t,x)∈R×Rand p≥2 is an integer.
Recall that (gKdV) admits a family of explicit traveling wave solutions Rc0,x0
indexed by (c0,x0)∈R∗
+×R.LetQbe the unique (up to translation) positive
solution in H1(R)(known also as ground state) to the following stationary elliptic
BXavier Friederich
friederich@math.unistra.fr
Raphaël Côte
cote@math.unistra.fr
1Institut de Recherche Mathématique Avancée UMR 7501, Université de Strasbourg, Strasbourg,
France
123
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