Modulation instability, conservation laws and localized waves for the generalized coupled Fokas-Lenells equation

TL;DR

This paper investigates the modulation instability, conservation laws, and localized wave solutions of the generalized coupled Fokas-Lenells equation, providing new analytical solutions and insights into rogue wave formation and interactions.

Contribution

It introduces a comprehensive analysis of the generalized coupled Fokas-Lenells equation, including stability analysis, conservation laws, and explicit localized wave solutions using Darboux transformation.

Findings

Distribution pattern of modulation instability gain G in the (K,k) plane.

Explicit determinant formulas for higher-order rogue waves.

Parameter-controlled interaction solutions and rogue wave degeneracy.

Abstract

This paper focuses on the modulation instability, conservation laws and localized wave solutions of the generalized coupled Fokas-Lenells equation. Based on the theory of linear stability analysis, distribution pattern of modulation instability gain G in the (K,k) frequency plane is depicted, and the constraints for the existence of rogue waves are derived. Subsequently, we construct the infinitely many conservation laws for the generalized coupled Fokas-Lenells equation from the Riccati-type formulas of the Lax pair. In addition, the compact determinant expressions of the N-order localized wave solutions are given via generalized Darboux transformation, including higher-order rogue waves and interaction solutions among rogue waves with bright-dark solitons or breathers. These solutions are parameter controllable: (m_i,n_i) and (\alpha,\beta) control the structure and ridge deflection of solution respectively, while the value of |d| controls the strength of interaction to realize energy exchange. Especially, when d=0, the interaction solutions degenerate into the corresponding order of rogue waves.