Generalized Darboux transformation and higher-order rogue wave solutions to the Manakov system

TL;DR

This paper develops a generalized Darboux transformation for the Manakov system, enabling the explicit construction of higher-order vector rogue wave solutions and analyzing their parametric dependencies.

Contribution

It introduces a recursive Darboux transformation method for the Manakov system to generate higher-order rogue wave solutions with a unified spectral parameter.

Findings

Constructed explicit first, second, and third-order rogue wave solutions.

Demonstrated that higher-order rogue waves depend on free parameters.

Provided visualizations of rogue wave features.

Abstract

In this paper, we construct a generalized recursive Darboux transformation of a focusing vector nonlinear Schr\"odinger equation known as the Manakov system. We apply this generalized recursive Darboux transformation to the Lax-pairs of this system in view of generating the Nth-order vector generalization rogue wave solutions with the same spectral parameter through a direct iteration rule. As a result, we discuss the first, second and third-order vector generalization rogue wave solutions while illustrating these features with some depictions. We show that higher-order rogue wave solutions depend on the values of their free parameters.