Heun-Wronskian Analysis of the Lax Spectrum for Sine-Gordon Kink-Breather Solutions

Year 2025, Issue: 53, 96 - 111, 31.12.2025

Volkan Ala

Abstract

We study the Lax spectrum of kink-breather solutions on cnoidal backgrounds for the integrable sine-Gordon equation. Linearizing around a kink-breather configuration on a periodic (cnoidal) carrier leads to a Schrödinger-type spectral problem with an elliptic potential given by the cosine of the background field. Using a Jacobi-elliptic change of variables adapted to the cnoidal structure, we reduce this second-order equation to a Heun-type differential equation. Two linearly independent Heun solutions are then used to build a Wronskian determinant whose zeros describe the Floquet-Bloch band-gap structure of the Lax spectrum. We discuss how the spectral bands and isolated eigenvalues (internal modes) depend on the physical and background parameters, and we relate limiting regimes such as the pure cnoidal background and the solitary kink limit to classical Lame- and Pöschl-Teller-type spectral problems.

Keywords

Heun equation , cnoidal waves , internal modes

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