A vectorial Darboux transformation for integrable matrix versions of the Fokas-Lenells equation

TL;DR

This paper develops a vectorial Darboux transformation for an integrable matrix form of the Fokas-Lenells equation, enabling systematic construction of various exact solutions including solitons and rogue waves.

Contribution

It introduces a novel vectorial binary Darboux transformation for matrix integrable systems and applies it to derive explicit solutions of the matrix Fokas-Lenells equation.

Findings

Derived matrix soliton solutions from trivial seed

Constructed breathers, dark solitons, rogue waves, and beating solitons

Systematic method for exact solutions on plane wave backgrounds

Abstract

Using bidifferential calculus, we derive a vectorial binary Darboux transformation for an integrable matrix version of the first negative flow of the Kaup-Newell hierarchy. A reduction from the latter system to an integrable matrix version of the Fokas-Lenells equation is then shown to inherit a corresponding vectorial Darboux transformation. Matrix soliton solutions are derived from the trivial seed solution. Furthermore, the Darboux transformation is exploited to determine in a systematic way exact solutions of the two-component vector Fokas-Lenells equation on a plane wave background. This comprises breathers, dark solitons, rogue waves and "beating solitons".