NLS breathers, rogue waves, and solutions of the Lyapunov equation for Jordan blocks

TL;DR

This paper derives infinite families of breather solutions to the focusing NLS equation using a matrix Darboux transformation with Jordan block spectral matrices, linking their structure to solutions of the Lyapunov equation.

Contribution

It introduces a novel matrix-based approach to generate NLS breathers via Jordan block spectral matrices and connects their properties to Lyapunov equation solutions.

Findings

Explicit families of Peregrine, Akhmediev, and Kuznetsov-Ma breathers derived.

Regularity of solutions established through Lyapunov equation properties.

Matrix Darboux transformation framework developed for NLS solutions.

Abstract

The infinite families of Peregrine, Akhmediev and Kuznetsov-Ma breather solutions of the focusing Nonlinear Schroedinger (NLS) equation are obtained via a matrix version of the Darboux transformation, with a spectral matrix of the form of a Jordan block. The structure of these solutions is essentially determined by the corresponding solution of the Lyapunov equation. In particular, regularity follows from properties of the Lyapunov equation.