Fokas-Lenells equation dark soliton and gauge equivalent spin equation

TL;DR

This paper develops a simplified Hirota bilinear method for the Fokas-Lenells equation to obtain dark soliton solutions and demonstrates its gauge equivalence to the Landau-Lifshitz spin system.

Contribution

It introduces an auxiliary function in bilinearization for the Fokas-Lenells equation, simplifying the process of finding dark soliton solutions and establishes gauge equivalence to a spin system.

Findings

Derived dark soliton solutions using a new bilinear method

Established gauge equivalence to the Landau-Lifshitz equation

Presented a Lax pair for the Fokas-Lenells equation

Abstract

We propose the Hirota bilinearization of the Fokas-Lenells derivative nonlinear Schrodinger equation with a non-vanishing background. The bilinear method is applied using an auxilary function to obtain the dark one soliton solution, dark two soliton solution and eventually the scheme for obtaining dark N soliton solutions. The use of auxilary function in bilinearization makes the method simpler than the ones reported earlier. Later, we have introduced a Lax pair for this integrable equation and using a transformation we have shown that this system is gauge equivalent to a spin system, namely the Landau-Lifshitz equation.