The Fokas-Lenells equations: Bilinear approach

TL;DR

This paper develops a bilinear approach to solve the Fokas-Lenells equations, deriving new solutions including multi-periodic and solitary waves, and connects these solutions to the two-dimensional massive Thirring model.

Contribution

It introduces a bilinear method for the Fokas-Lenells equations, deriving novel solutions related to real discrete eigenvalues and linking to the Thirring model.

Findings

Derived double Wronskian solutions for Fokas-Lenells equations

Obtained new solutions with real discrete eigenvalues

Connected solutions to the two-dimensional massive Thirring model

Abstract

In this paper, the Fokas-Lenells equations are investigated via bilinear approach. We bilinearize the unreduced Fokas-Lenells system, derive double Wronskian solutions, and then, by means of a reduction technique we obtain variety of solutions of the reduced equations. This enables us to have a full profile of solutions of the classical and nonlocal Fokas-Lenells equations. Some obtained solutions are illustrated based on asymptotic analysis. As a notable new result, we obtain solutions to the Fokas-Lenells equation, which are related to real discrete eigenvalues and not reported before in the analytic approaches. These solutions behave like (multi-)periodic waves or solitary waves with algebraic decay. In addition, we also obtain solutions to the two-dimensional massive Thirring model from those of the Fokas-Lenells equation.