Invariant manifolds around soliton manifolds for the nonlinear Klein-Gordon equation

TL;DR

This paper constructs invariant manifolds around soliton solutions for the nonlinear Klein-Gordon equation using a graph transform method that handles symmetries with fewer spectral assumptions.

Contribution

It introduces a graph transform approach to build invariant manifolds for the nonlinear Klein-Gordon equation, accommodating modulation parameters and requiring less spectral information.

Findings

Successfully constructs center-stable and center-unstable manifolds.

Method reduces spectral assumptions compared to previous approaches.

Applicable to a family of solitary waves generated by symmetries.

Abstract

We construct center-stable and center-unstable manifolds, as well as stable and unstable manifolds, for the nonlinear Klein-Gordon equation with a focusing energy sub-critical nonlinearity, associated with a family of solitary waves which is generated from any radial stationary solution by the action of all Lorentz transforms and spatial translations. The construction is based on the graph transform (or Hadamard) approach, which requires less spectral information on the linearized operator, and less decay of the nonlinearity, than the Lyapunov-Perron method employed previously in this context. The only assumption on the stationary solution is that the kernel of the linearized operator is spanned by its spatial derivatives, which is known to hold for the ground states. The main novelty of this paper lies with the fact that the graph transform method is carried out in the presence of modulation parameters corresponding to the symmetries.