Nonlinear Schr\"odinger Equation: Generalized Darboux Transformation and Rogue Wave Solutions

TL;DR

This paper develops a generalized Darboux transformation for the nonlinear Schrödinger equation, enabling explicit construction of higher-order rogue wave solutions and analyzing their complex dynamics.

Contribution

It introduces a new generalized Darboux transformation framework and provides compact formulas for rogue wave solutions of the nonlinear Schrödinger and Hirota equations.

Findings

Explicit formulas for N-th order rogue waves

Analysis of third order rogue wave dynamics

Unified approach for different integrable equations

Abstract

In this paper, we construct a generalized Darboux transformation for nonlinear Schr\"odinger equation. The associated $N$-fold Darboux transformation is given both in terms of a summation formula and in terms of determinants. As applications, we obtain compact representations for the $N$-th order rogue wave solutions of the focusing nonlinear Schr\"odinger equation and Hirota equation. In particular, the dynamics of the general third order rogue wave is discussed and shown to exhibit interesting structure.