GBDT with nontrivial seeds: explicit solutions of the focusing NLS equations and the corresponding Weyl functions

TL;DR

This paper develops explicit solutions for the focusing nonlinear Schr"odinger equation using a GBDT approach with nontrivial exponential seeds, providing insights into rogue waves and modulation phenomena.

Contribution

It introduces a GBDT method with explicit solutions for NLS equations using exponential seeds, extending previous trivial seed work.

Findings

Explicit solutions for focusing NLS with exponential seeds

Derivation of Baker-Akhiezer functions and Weyl function evolution

Application to rogue waves and modulation solutions

Abstract

Our GBDT (generalised B\"acklund-Darboux transformation) approach is used to construct explicit solutions of the focusing nonlinear Schr\"odinger (NLS) equation in the case of the exponential seed $a \exp\{2 i (cx +dt)\}$. The corresponding Baker-Akhiezer functions and evolution of the Weyl functions are obtained as well. In particular, the solutions, which appear in the study of rogue waves, step-like solutions and $N$-modulation solutions of the NLS equation are considered. This work is an essential development of our joint work with Rien Kaashoek and Israel Gohberg, where the seed was trivial, as well as several other of our previous works.