# ON EXISTENCE AND UNIQUENESS OF ASYMPTOTIC $N$-SOLITON-LIKE SOLUTIONS OF THE NONLINEAR KLEIN-GORDON EQUATION

Abstract : We are interested in solutions of the nonlinear Klein-Gordon equation (NLKG) in $\mathbb{R}^{1+d}$, $d\ge1$, 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 given (unstable) solitons. For $N = 1$, this family completely describes the set of solutions converging to the soliton considered; for $N\ge 2$, we prove uniqueness in a class with explicit algebraic rate of convergence.
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https://hal.archives-ouvertes.fr/hal-03244342
Contributor : Xavier Friederich <>
Submitted on : Wednesday, June 16, 2021 - 7:53:46 AM
Last modification on : Friday, June 18, 2021 - 3:36:58 AM

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• HAL Id : hal-03244342, version 2
• ARXIV : 2106.01106

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Xavier Friederich. ON EXISTENCE AND UNIQUENESS OF ASYMPTOTIC $N$-SOLITON-LIKE SOLUTIONS OF THE NONLINEAR KLEIN-GORDON EQUATION. 2021. ⟨hal-03244342v2⟩

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