On existence and uniqueness of asymptotic $N$-soliton-like solutions of the nonlinear klein-gordon equation

Abstract

A. We are interested in solutions of the nonlinear Klein-Gordon equation (NLKG) in R 1+ , ≥ 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 -parameter family of solutions of (NLKG) which converges exponentially fast to a sum of given (unstable) solitons. For = 1, this family completely describes the set of solutions converging to the soliton considered; for ≥ 2, we prove uniqueness in a class with explicit algebraic rate of convergence.