Generalized perturbation (n, M)-fold Darboux transformations and multi-rogue-wave structures for the modified self-steepening nonlinear Schrodinger equation

TL;DR

This paper develops a new method for constructing multi-rogue-wave solutions of the modified nonlinear Schrödinger equation using generalized Darboux transformations, revealing complex wave structures and potential applications in optics.

Contribution

It introduces a simple, constructive approach to derive generalized perturbation (n,M)-fold Darboux transformations for the MNLS equation, enabling explicit multi-rogue-wave solutions and analysis.

Findings

Explicit multi-rogue-wave solutions with complex structures like triangles and pentagons.

Numerical simulations illustrating the dynamical behaviors of rogue waves.

Extension of the method to other integrable equations like the Gerjikov-Ivanov equation.

Abstract

In this paper, a simple and constructive method is presented to find the generalized perturbation (n,M)-fold Darboux transformations (DTs) of the modified nonlinear Schrodinger (MNLS) equation in terms of fractional forms of determinants. In particular, we apply the generalized perturbation (1,N-1)-fold DTs to find its explicit multi-rogue-wave solutions. The wave structures of these rogue-wave solutions of the MNLS equation are discussed in detail for different parameters, which display abundant interesting wave structures, including the triangle and pentagon, etc. and may be useful to study the physical mechanism of multirogue waves in optics. The dynamical behaviors of these multi-rogue-wave solutions are illustrated using numerical simulations. The same Darboux matrix can also be used to investigate the Gerjikov-Ivanov equation such that its multi-rogue-wave solutions and their wave structures are also found. The method can also be extended to find multi-rogue-wave solutions of other nonlinear integrable equations.