General high-order rogue waves and their dynamics in the nonlinear Schroedinger equation

TL;DR

This paper derives general high-order rogue wave solutions for the nonlinear Schrödinger equation using the bilinear method, revealing their parameter dependence and diverse dynamic patterns beyond previously known solutions.

Contribution

It introduces a determinant-based formulation of high-order rogue waves with free parameters, expanding understanding of their structure and dynamics in the nonlinear Schrödinger equation.

Findings

Derived explicit formulas for high-order rogue waves

Identified free parameters controlling wave patterns

Demonstrated diverse solution dynamics including wave arrays

Abstract

General high-order rogue waves in the nonlinear Schroedinger equation are derived by the bilinear method. These rogue waves are given in terms of determinants whose matrix elements have simple algebraic expressions. It is shown that the general N-th order rogue waves contain N-1 free irreducible complex parameters. In addition, the specific rogue waves obtained by Akhmediev et al. (Phys. Rev. E 80, 026601 (2009)) correspond to special choices of these free parameters, and they have the highest peak amplitudes among all rogue waves of the same order. If other values of these free parameters are taken, however, these general rogue waves can exhibit other solution dynamics such as arrays of fundamental rogue waves arising at different times and spatial positions and forming interesting patterns.