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

Abstract

This paper is devoted to the analysis of the following nonlinear wave equation \[ u_{tt} - u_{xx} + (1 + qδ_0(x)) \sin u = 0, \] 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 $H_{\sin }^1\times L^2$. 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.

24 pages, 1 figure