Stability of Standing Waves for a Nonlinear Klein-Gordon Equation with Delta Potentials

TL;DR

This paper investigates the local well-posedness and orbital stability of standing waves in a nonlinear Klein-Gordon equation with delta potentials, highlighting spectral analysis and Hamiltonian structure.

Contribution

It establishes the well-posedness and stability results for standing waves in a singularly perturbed Klein-Gordon equation with delta potentials, a novel setting.

Findings

Local well-posedness via fixed point argument

Spectral analysis of linearized operator

Stability and instability results for standing waves

Abstract

In this paper, we study local well-posedness and orbital stability of standing waves for a singularly perturbed one-dimensional nonlinear Klein-Gordon equation. We first establish local well-posedness of the Cauchy problem by a fixed point argument. Unlike the unperturbed case, a noteworthy difficulty here arises from the possible non-unitarity of the semigroup generating the corresponding linear evolution. We then show that the equation is Hamiltonian and we establish several stability/instability results for its standing waves. Our analysis relies on a detailed study of the spectral properties of the linearization of the equation, and on the well-known 'slope condition' for orbital stability.