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

Xavier Friederich (1)

1 IRMA - Institut de Recherche Mathématique Avancée

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.

Mots clés: nonlinear Klein-Gordon equation; solitons; multi-solitons; classification

Domaines: Equations aux dérivées partielles [math.AP]

Xavier Friederich. ON EXISTENCE AND UNIQUENESS OF ASYMPTOTIC $N$-SOLITON-LIKE SOLUTIONS OF THE NONLINEAR KLEIN-GORDON EQUATION. 2021. ⟨hal-03244342v1⟩