The Fokas-Lenells equation on the line: Global well-posedness with solitons

TL;DR

This paper establishes the global existence of solutions to the Fokas-Lenells equation on the line with solitons, using a modified Darboux transformation and inverse scattering transform techniques.

Contribution

It introduces a new modified Darboux transformation to handle solitons and proves global well-posedness for initial data including solitons.

Findings

Global solutions exist in specified Sobolev spaces.

The modified Darboux transformation effectively adds or removes solitons.

The inverse scattering transform is used to establish well-posedness with finite solitons.

Abstract

In this paper, we prove the existence of global solutions in $H^3(\mathbb{R})\cap H^{2,1}(\mathbb{R})$ to the Fokas-Lenells (FL) equation on the line when the initial data includes solitons.A key tool in proving this result is a newly modified Darboux transformation, which adds or subtracts a soliton with given spectral and scattering parameters. In this way the inverse scattering transform technique is then applied to establish the global well-posedness of initial value problem with a finite number of solitons based on our previous results on the global well-posedness of the FL equation.