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

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.