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On existence and uniqueness of asymptotic N-soliton-like solutions of the nonlinear Klein–Gordon equation

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Xavier Friederich
Xavier Friederich
Xavier Friederich
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Abstract

We are interested in solutions of the nonlinear Klein–Gordon equation (NLKG) in R1+dR1+d\mathbb {R}^{1+d}, d≥1d≥1d\ge 1, which behave as a soliton or a sum of solitons in large time. In the spirit of other articles focusing on the supercritical generalized Korteweg-de Vries equations and on the nonlinear Schrödinger equations, we obtain an N-parameter family of solutions of (NLKG) which converges exponentially fast to a sum of N given (unstable) solitons. For N=1N=1, this family completely describes the set of solutions converging to the soliton considered; for N≥2N≥2N\ge 2, we prove uniqueness in a class with explicit algebraic rate of convergence.
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Mathematische Zeitschrift (2022) 302:2131–2191
https://doi.org/10.1007/s00209-022-03137-x
Mathematische Zeitschrift
On existence and uniqueness of asymptotic N-soliton-like
solutions of the nonlinear Klein–Gordon equation
Xavier Friederich1
Received: 21 June 2021 / Accepted: 26 August 2022 / Published online: 21 September 2022
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022
Abstract
We are interested in solutions of the nonlinear Klein–Gordon equation (NLKG) in R1+d,
d≥1, which behave as a soliton or a sum of solitons in large time. In the spirit of other
articles focusing on the supercritical generalized Korteweg-de Vries equations and on the
nonlinear Schrödinger equations, we obtain an N-parameter family of solutions of (NLKG)
which converges exponentially fast to a sum of Ngiven (unstable) solitons. For N=1, this
family completely describes the set of solutions converging to the soliton considered; for
N≥2, we prove uniqueness in a class with explicit algebraic rate of convergence.
Keywords Nonlinear Klein–Gordon equation ·Solitons ·Multi-solitons ·Classification
Mathematics Subject Classification Primary 35Q51 ·35L71; Secondary 35B40 ·35C08 ·
37K40
1 Introduction
1.1 Setting of the problem
We consider the following nonlinear Klein–Gordon equation
∂2
tu=u−u+f(u), (NLKG)
where uis a real-valued function of (t,x)∈R×Rdand fis a C1real-valued function on
R. This equation classically rewrites as the following first order system in time:
∂tU=0Id
−Id 0U+0
f(u),(NLKG’)
BXavier Friederich
friederich@math.unistra.fr
1Institut de Recherche Mathématique Avancée UMR 7501, Université de Strasbourg, Strasbourg, France
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... With particular non-linearities obtained by linear combinations of quadratic, cubic, and quintic terms it appears also when one tries to describe the propagation of laser beams in some mediums [4] or of more general ultrashort optical pulses (see for example [22] for the study of some solutions of these equations), with applications in medical imaging, material processing and optical communications (we refer to [29] for further Ce chapitre fait l'objet d'un article rédigé en collaboration avec Raphaël Côte et accepté pour publication dans Communications in Partial Differential Equations [16]. 31 details concerning the applications of (NLS) to fiber optics for example). ...
... Ce chapitre fait l'objet d'un article soumis pour publication [31]. ...
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