Linear and orbital stability analysis for solitary-wave solutions of variable-coefficient scalar field equations

TL;DR

This paper investigates how small, slowly decaying variable coefficients in scalar field equations affect the stability of kink solutions, providing spectral analysis and conditions for eigenvalue emergence, and establishing orbital stability despite potential negative eigenvalues.

Contribution

It introduces a method to analyze the spectral stability of kink solutions under variable coefficients and proves their orbital stability in this perturbed setting.

Findings

Existence of stationary kink solutions in variable-coefficient equations

Derivation of a formula to detect eigenvalue emergence from the essential spectrum

Orbital stability of solitary waves despite negative eigenvalues

Abstract

We study general semilinear scalar-field equations on the real line with variable coefficients in the linear terms. These coefficients are uniformly small, but slowly decaying, perturbations of a constant-coefficient operator. We are motivated by the question of how these perturbations of the equation may change the stability properties of kink solutions (one-dimensional topological solitons). We prove existence of a stationary kink solution in our setting, and perform a detailed spectral analysis of the corresponding linearized operator, based on perturbing the linearized operator around the constant-coefficient kink. We derive a formula that allows us to check whether a discrete eigenvalue emerges from the essential spectrum under this perturbation. Known examples suggest that this extra eigenvalue may have an important influence on the long-time dynamics in a neighborhood of the kink. We also establish orbital stability of solitary-wave solutions in the variable-coefficient regime, despite the possible presence of negative eigenvalues in the linearization.