Stability of elliptic solutions to the sinh-Gordon equation

TL;DR

This paper proves the spectral and orbital stability of elliptic solutions to the sinh-Gordon equation by leveraging its integrability and conserved quantities, demonstrating robustness against subharmonic perturbations.

Contribution

It introduces a Lyapunov functional based on higher-order conserved quantities to establish stability of elliptic solutions, a novel approach in this context.

Findings

Elliptic solutions are spectrally stable.

Elliptic solutions are orbitally stable under subharmonic perturbations.

Stability proof utilizes conserved quantities and Lyapunov functional.

Abstract

Using the integrability of the sinh-Gordon equation, we demonstrate the spectral stability of its elliptic solutions. By constructing a Lyapunov functional using higher-order conserved quantities of the sinh-Gordon equation, we show that these elliptic solutions are orbitally stable with respect to subharmonic perturbations of arbitrary period.