On the global well-posedness for the Fokas-Lenells equation on the line

We obtain the global well-posedness to the Cauchy problem of the Fokas-Lenells (FL) equation on the line without the small-norm assumption on initial data $u_0\in H^3\left(R\right)\cap H^{2,1}\left(R\right)$. Our main technical tool is the inverse scattering transform method based on the representation of a Riemann-Hilbert (RH) problem associated with the above Cauchy problem. The existence and the uniqueness of the RH problem is shown via a general vanishing lemma. By representing the solutions of the RH problem via the Cauchy integral protection and the reflection coefficients, the reconstruction formula is used to obtain a unique local solution of the FL equation. Further, the eigenfunctions and the reflection coefficients are shown Lipschitz continuous with respect to initial data, which provides a prior estimate of the solution to the FL equation. Based on the local solution and the uniformly prior estimate, we construct a unique global solution in $H^3\left(R\right)\cap H^{2,1}\left(R\right)$ to the FL equation.