Stability of standing periodic waves in the massive Thirring model

TL;DR

This paper investigates the spectral stability of standing periodic waves in the massive Thirring model, using spectral analysis and numerical methods to identify stability conditions based on eigenvalue locations.

Contribution

It provides a novel spectral stability criterion for standing waves in the massive Thirring model by linking eigenvalue positions to stability.

Findings

Standing waves are stable if eigenvalues are on the imaginary axis or diagonals.

Spectral stability depends on the eigenvalues coinciding with spectral band endpoints.

Analytical and numerical methods jointly determine stability regions.

Abstract

We analyze the spectral stability of the standing periodic waves in the massive Thirring model in laboratory coordinates. Since solutions of the linearized MTM equation are related to the squared eigenfunctions of the linear Lax system, the spectral stability of the standing periodic waves can be studied by using their Lax spectrum. Standing periodic waves are classified based on eight eigenvalues which coincide with the endpoints of the spectral bands of the Lax spectrum. Combining analytical and numerical methods, we show that the standing periodic waves are spectrally stable if and only if the eight eigenvalues are located either on the imaginary axis or along the diagonals of the complex plane.