Existence of global solutions to the Fokas-Lenells equation with arbitrary spectral singularities

TL;DR

This paper proves the global existence of solutions to the Fokas-Lenells equation for arbitrary initial data in specific Sobolev spaces, removing previous spectral restrictions using inverse scattering and Riemann-Hilbert techniques.

Contribution

It extends prior results by establishing global solutions without spectral restrictions, employing inverse scattering transform and Zhou's bijectivity theory.

Findings

Global solutions exist for any initial data in the specified Sobolev space.

Spectral restrictions on initial data are removed.

The proof utilizes Riemann-Hilbert problems and Zhou's theory.

Abstract

We establish the global existence of solutions to the Fokas-Lenells equation for any initial data in a weighted Sobolev space $H^{3}(\mathbb{R})\cap H^{2,1}(\mathbb{R})$.This result removes all spectral restrictions on the initial data required in our previous work. The proof primarily relies on the inverse scattering transform formulated as new Riemann-Hilbert problems and Zhou's $L^{2}$-Sobolev bijectivity theory.