The Riemann-Hilbert approach for the integrable fractional Fokas--Lenells equation

TL;DR

This paper introduces a new integrable fractional Fokas--Lenells equation, solves it using the Riemann-Hilbert method, and derives explicit multi-soliton solutions demonstrating their asymptotic behavior.

Contribution

It develops a novel integrable fractional equation and provides explicit soliton solutions using the Riemann-Hilbert approach, including a rigorous proof of the one-soliton case.

Findings

Explicit fractional N-soliton solutions in determinant form

Asymptotic linear superposition of solitons as time goes to infinity

Rigorous proof of the fractional one-soliton solution

Abstract

In this paper, we propose a new integrable fractional Fokas--Lenells equation by using the completeness of the squared eigenfunctions, dispersion relation, and inverse scattering transform. To solve this equation, we employ the Riemann-Hilbert approach. Specifically, we focus on the case of the reflectionless potential with a simple pole for the zero boundary condition. And we provide the fractional $N$-soliton solution in determinant form. Additionally, we prove the fractional one-soliton solution rigorously. Notably, we demonstrate that as $|t|\to\infty$, the fractional $N$-soliton solution can be expressed as a linear combination of $N$ fractional single-soliton solutions.