Stationary multi-kinks in the discrete sine-Gordon equation

TL;DR

This paper investigates the existence, spectral stability, and eigenvalue structure of static multi-kink solutions in the discrete sine-Gordon equation, combining analytical and numerical methods.

Contribution

It introduces a method to construct multi-kinks from well-separated kinks and antikinks, and derives explicit eigenvalue formulas linked to primary kink spectra.

Findings

Eigenvalues of multi-kinks are closely related to those of primary kinks.

Spectral stability is maintained when primary kink spectra are imaginary.

Numerical simulations confirm analytical spectral predictions.

Abstract

We consider the existence and spectral stability of static multi-kink structures in the discrete sine-Gordon equation, as a representative example of the family of discrete Klein-Gordon models. The multi-kinks are constructed using Lin's method from an alternating sequence of well-separated kink and antikink solutions. We then locate the point spectrum associated with these multi-kink solutions by reducing the spectral problem to a matrix equation. For an $m$-structure multi-kink, there will be $m$ eigenvalues in the point spectrum near each eigenvalue of the primary kink, and, as long as the spectrum of the primary kink is imaginary, the spectrum of the multi-kink will be as well. We obtain analytic expressions for the eigenvalues of a multi-kink in terms of the eigenvalues and corresponding eigenfunctions of the primary kink, and these are in very good agreement with numerical results. We also perform numerical time-stepping experiments on perturbations of multi-kinks, and the outcomes of these simulations are interpreted using the spectral results.