Unstable kink-soliton profiles for the sine-Gordon equation on a $\mathcal{Y}$-junction graph with $\delta$-interaction

TL;DR

This paper proves linear instability of kink and kink/anti-kink soliton solutions for the sine-Gordon equation on a Y-junction graph with delta-interaction boundary conditions, using spectral analysis and operator theory.

Contribution

It introduces a linear instability criterion for sine-Gordon solitons on metric graphs with delta interactions, extending stability analysis techniques to complex network structures.

Findings

Kink and kink/anti-kink profiles are linearly and nonlinearly unstable.

Develops a spectral instability criterion based on eigenvalues of the linearized operator.

Provides a well-posedness framework for the sine-Gordon model on Y-junction graphs.

Abstract

The aim of this work is to establish a linear instability result of stationary, kink and kink/anti-kink soliton profile solutions for the sine-Gordon equation on a metric graph with a structure represented by a $\mathcal Y$-junction. The model considers boundary conditions at the graph-vertex of $\delta$-interaction type, or in other words, continuity of the wave functions at the vertex plus a law of Kirchhoff-type for the flux. It is shown that kink and kink/anti-kink soliton type stationary profiles are linearly (and nonlinearly) unstable. For that purpose, a linear instability criterion that provides the sufficient conditions on the linearized operator around the wave to have a pair of real positive/negative eigenvalues, is established. As a result, the linear stability analysis depends upon of the spectral study of this linear operator and of its Morse index. The extension theory of symmetric operators, Sturm-Liouville oscillation results and analytic perturbation theory of operators are fundamental ingredients in the stability analysis. A comprehensive study of the local well-posedness of the sine-Gordon model in $\mathcal E(\mathcal Y) \times L^2(\mathcal{Y})$ where $\mathcal E(\mathcal Y) \subset H^1(\mathcal{Y})$ is an appropriate energy space, is also established. The theory developed in this investigation has prospects for the study of the instability of stationary wave solutions of other nonlinear evolution equations on metric graphs.