Stability of Breathers for a Periodic Klein-Gordon Equation

TL;DR

This paper investigates the existence and stability of breather solutions in a periodic Klein-Gordon equation, combining analytical and numerical methods to construct and analyze these localized, time-periodic waves.

Contribution

It introduces a numerical approach to construct breather solutions with high accuracy and explores their stability in a spatially heterogeneous $^4$ model, revealing their generic instability.

Findings

Breather solutions are generally unstable in the studied model.

Numerical methods effectively construct breathers with desired accuracy.

Instability leads to motion of the breather structures.

Abstract

The existence of breather type solutions, i.e., periodic in time, exponentially localized in space solutions, is a very unusual feature for continuum, nonlinear wave type equations. Following an earlier work [Comm. Math. Phys. {\bf 302}, 815-841 (2011)], establishing a theorem for the existence of such structures, we bring to bear a combination of analysis-inspired numerical tools that permit the construction of such wave forms to a desired numerical accuracy. In addition, this enables us to explore their numerical stability. Our computations show that for the spatially heterogeneous form of the $\phi^4$ model considered herein, the breather solutions are generically found to be unstable. Their instability seems to generically favor the motion of the relevant structures. We expect that these results may inspire further studies towards the identification of stable continuous breathers in spatially-heterogeneous, continuum nonlinear wave equation models.