Modulational instability of periodic standing waves in the derivative NLS equation

TL;DR

This paper classifies all periodic standing waves in the derivative nonlinear Schrödinger equation using an algebraic method and analyzes their modulational instability through spectral band approximation.

Contribution

It introduces a new algebraic approach with two eigenvalues to classify periodic standing waves in the DNLS equation, linking spectral properties to stability.

Findings

Complete classification of periodic standing waves

Spectral analysis of modulational instability

Numerical approximation of spectral bands

Abstract

We consider the periodic standing waves in the derivative nonlinear Schrodinger (DNLS) equation arising in plasma physics. By using a newly developed algebraic method with two eigenvalues, we classify all periodic standing waves in terms of eight eigenvalues of the Kaup-Newell spectral problem located at the end points of the spectral bands outside the real line. The analytical work is complemented with the numerical approximation of the spectral bands, this enables us to fully characterize the modulational instability of the periodic standing waves in the DNLS equation.