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

TL;DR

This work proves global well-posedness for the Fokas-Lenells equation with arbitrary spectral data by leveraging an inverse scattering framework built around Riemann-Hilbert problems. A novel N(x,k) construction removes eigenvalues and spectral singularities, and a KN-to-ZS transformation recasts the problem on the z-plane, enabling Type I and II vector RH problems to recover the potential and its derivatives via Beals-Coifman methods. Time evolution of the jump, together with Zhou's $L^{2}$-Sobolev bijectivity theory, yields global existence for initial data in $H^{3}()\cap H^{2,1}()$ with Lipschitz dependence. The approach also introduces two reconstruction formulas to obtain $u$, $u_x$, and $u_{xx}$, ensuring the required regularity $u\in H^{3}\cap H^{2,1}$ for all times, thereby extending global well-posedness results for the FL equation beyond prior spectral restrictions.

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.

Figures (6)

• Figure 1: Jump contour $\Gamma_{0}$ and analytic domains $D^{\pm}$ for $n(x,k)$.
• Figure 2: Jump contour $\Gamma$ and analytic domains $\widetilde{D}^{\pm}$ for $N(x,k)$.
• Figure 3: Jump contour $\Lambda$ and analytic domains $\Omega^{\pm}$ for $P(x,z)$.
• Figure 4: Jump contour $\widetilde{\Lambda}$ and analytic domains $\widetilde{\Omega}^{\pm}$ for $P^{(2)}(x,z)$.
• Figure 5: The relationships among RH problems.

Theorems & Definitions (14)

• Theorem 1.1
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• Lemma 3.4
• proof
• Proposition 3.5
• proof