Bilinearization of the Fokas-Lenells equation Conservation laws and soliton interactions

TL;DR

This paper introduces a simplified bilinearization method for the Fokas-Lenells equation, deriving explicit soliton solutions and analyzing their elastic interactions, along with a generalized Lax pair and conserved quantities.

Contribution

It presents a novel bilinearization scheme using an auxiliary function, producing more general soliton solutions and explicit interaction analysis for the Fokas-Lenells equation.

Findings

Derived explicit 1- and 2-soliton solutions

Confirmed elastic soliton interactions with phase shifts

Proposed a generalized Lax pair and conserved quantities

Abstract

In this paper, we propose the bilinearization of the Fokas-Lenells equation (FLE) with a vanishing boundary condition. In the proposed bilinearization we make use of an auxiliary function to convert the trilinear equations into a set of bilinear equations. We obtain bright 1-soliton, 2- soliton solutions and present the scheme for obtaining N soliton solution. In the soliton solution the presence of an additional parameter allows tuning the position of soliton. We find that the proposed scheme of bilinearization using auxiliary function, considerably simplifies the procedure yet generates a more general solution than the one reported earlier. We show that the obtained soliton solution reduces to an algebraic soliton in the limit of infinite width. Further we show explicitly that the soliton interactions are elastic through asymptotic analysis, that is the amplitude of each soliton remains same before and after interaction. The mark of interaction is left behind only in the phase of each soliton. Secondly, we propose a generalised Lax pair for the FLE and obtain the conserved quantities by solving Riccati equation. We believe that the present investigation would be useful to study the applications of FLE in nonlinear optics and other branches of physics.