Generalized Darboux transformation and localized waves in coupled Hirota equations

TL;DR

This paper develops a generalized Darboux transformation for coupled Hirota equations with high-order nonlinear effects, deriving various localized wave solutions and revealing complex dynamic structures in coupled systems.

Contribution

It introduces a novel generalized Darboux transformation for coupled Hirota equations, enabling the construction of high-order localized wave solutions with complex interactions.

Findings

Derived Nth-order localized wave solutions on plane backgrounds.

Obtained semi-rational, multi-parametric localized wave solutions.

Revealed complex interaction dynamics of localized waves.

Abstract

In this paper, we construct a generalized Darboux transformation to the coupled Hirota equations with high-order nonlinear effects like the third dispersion, self-steepening and inelastic Raman scattering terms. As application, an Nth-order localized wave solution on the plane backgrounds with the same spectral parameter is derived through the direct iterative rule. In particular, some semi-rational, multi-parametric localized wave solutions are obtained: (1) Vector generalization of the first- and the second-order rogue wave solution; (2) Interactional solutions between a dark-bright soliton and a rogue wave, two dark-bright solitons and a second-order rogue wave; (3) Interactional solutions between a breather and a rogue wave, two breathers and a second-order rogue wave. The results further reveal the striking dynamic structures of localized waves in complex coupled systems.