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Multi-solitons for nonlinear Klein–Gordon equations

Part of: Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  09 June 2014

RAPHAËL CÔTE  and
CLAUDIO MUÑOZ
RAPHAËL CÔTE
Affiliation:
CNRS and École polytechnique, Centre de Mathématiques Laurent Schwartz UMR 7640, Route de Palaiseau, 91128 Palaiseau cedex, France; cote@math.polytechnique.fr
CLAUDIO MUÑOZ
Affiliation:
Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, IL 60615, U.S.A.; cmunoz@math.uchicago.edu

Abstract

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In this paper, we consider the existence of multi-soliton structures for the nonlinear Klein–Gordon (NLKG) equation in $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{R}^{1+d}$. We prove that, independently of the unstable character of NLKG solitons, it is possible to construct a $N$-soliton family of solutions to the NLKG equation, of dimension $2N$, globally well defined in the energy space $H^1\times L^2$ for all large positive times. The method of proof involves the generalization of previous works on supercritical Nonlinear Schrödinger (NLS) and generalized Korteweg–de Vries (gKdV) equations by Martel, Merle, and the first author [R. Côte, Y. Martel and F. Merle, Rev. Mat. Iberoam. 27 (1) (2011), 273–302] to the wave case, where we replace the unstable mode associated to the linear NLKG operator by two generalized directions that are controlled without appealing to modulation theory. As a byproduct, we generalize the linear theory described in Grillakis, Shatah, and Strauss [J. Funct. Anal. 74 (1) (1987), 160–197] and Duyckaerts and Merle [Int. Math. Res. Pap. IMRP (2008), Art ID rpn002] to the case of boosted solitons, and provide new solutions to be studied using the recent work of Nakanishi and Schlag [Zurich Lectures in Advanced Mathematics, vi+253 pp (European Mathematical Society (EMS), Zürich, 2011)] theory.

MSC classification

Secondary: 35Q51: Soliton-like equations 35Q40: PDEs in connection with quantum mechanics
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e15
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2014

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