Analysis of the sine-Gordon equation with a nonlinear $\delta$-potential

TL;DR

This paper analyzes a nonlinear wave equation with a delta potential, establishing global well-posedness, characterizing stationary solutions, and determining their stability based on the potential parameter.

Contribution

It provides a comprehensive analysis of the nonlinear sine-Gordon equation with a delta potential, including well-posedness, stationary wave classification, and stability criteria.

Findings

Global well-posedness in the energy space is proved.

Complete characterization of stationary waves based on parameter q.

Stability or instability of stationary waves depends on the sign of q.

Abstract

This paper is devoted to the analysis of the following nonlinear wave equation \[ u_{tt} - u_{xx} + (1 + q\delta_0(x)) \sin u = 0, \] where $\delta_0 = \delta_0(x)$ is the Dirac delta function centered at the origin and $q \in \mathbb{R}$ is a constant. Equations of this form arise in the study of propagating solitons in the presence of a localized inhomogeneity. It is proved that the Cauchy problem for this equation is globally well-posed in the energy space $H^1_{\sin} \times L^2$. A complete characterization of stationary waves in the energy space, based on the parameter $q$, is also provided. Finally, a criterion to determine the stability or instability of the stationary waves, which depends upon the sign of the parameter $q$, is established.