Generalized Darboux transformation and N-th order rogue wave solution of a general coupled nonlinear Schr\"{o}dinger equations

TL;DR

This paper develops a generalized Darboux transformation for a coupled nonlinear Schrödinger system, enabling explicit construction of higher-order rogue wave solutions and revealing diverse wave pattern formations.

Contribution

It introduces a recursive GDT method and determinant formulas for N-th order rogue waves in a coupled NLS system, with new pattern formation insights.

Findings

Derived explicit first, second, and third order rogue wave solutions.

Demonstrated formation of triplet, triangle, and hexagonal rogue wave patterns.

Provided a flexible framework for analyzing complex wave interactions.

Abstract

We construct a generalized Darboux transformation (GDT) of a general coupled nonlinear Schr\"{o}dinger (GCNLS) system. Using GDT method we derive a recursive formula and present determinant representations for N-th order rogue wave solution of this system. Using these representations we derive first, second and third order rogue wave solutions with certain free parameters. By varying these free parameters we demonstrate the formation of triplet, triangle and hexagonal patterns of rogue waves.