Existence and multiplicity of stable bound states for the nonlinear Klein-Gordon equation

TL;DR

This paper investigates the conditions under which soliton-like solutions exist and are multiple for the nonlinear Klein-Gordon equation, emphasizing verifiable criteria and the influence of the nonlinear term's shape.

Contribution

It provides necessary and sufficient conditions for soliton existence and demonstrates that soliton multiplicity depends on the nonlinear term's shape, independent of domain topology.

Findings

Conditions for soliton existence are easily verified.

Multiplicity of solitons is guaranteed by the nonlinear term's shape.

Results apply to equations on ^N without domain topological considerations.

Abstract

We are interested in the problem of existence of soliton-like solutions for the nonlinear Klein-Gordon equation. In particular we study some necessary and sufficient conditions on the nonlinear term to obtain solitons of a given charge. We remark that the conditions we consider can be easily verified. Moreover we show that multiplicity of solitons of the same charge is guaranteed by the "shape" of the nonlinear term for equations on $\R^{N}$, hence without appealing to topological or geometrical properties of the domain.