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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 ∈ H 3 (R) ∩ H 2 , 1 (R). 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 (R) ∩ H 2 , 1 (R) to the FL equation. [ABSTRACT FROM AUTHOR]

We study the long-time asymptotic behavior of the focusing Fokas-Lenells (FL) equation u x t + α β 2 u − 2 i α β u x − α u x x − i α β 2 | u | 2 u x = 0 with generic initial data in a Sobolev space which supports bright soliton solutions. The FL equation is an integrable generalization of the well-known Schrodinger equation, and also linked to the derivative Schrodinger model, but it exhibits several different characteristics from theirs. (i) The Lax pair of the FL equation involves an additional spectral singularity at k = 0. (ii) Four stationary phase points will appear during asymptotic analysis, which require a more detailed necessary description to obtain the long-time asymptotics of the focusing FL equation. Based on the Riemann-Hilbert problem for the initial value problem of the focusing FL equation, we show that inside any fixed time-spatial cone C (x 1 , x 2 , v 1 , v 2) = { (x , t) ∈ R 2 | x = x 0 + v t , x 0 ∈ [ x 1 , x 2 ] , v ∈ [ v 1 , v 2 ] } , the long-time asymptotic behavior of the solution u (x , t) for the focusing FL equation can be characterized with an N (I) -soliton on discrete spectrums and a leading order term O (t − 1 / 2) on continuous spectrum up to a residual error order O (t − 3 / 4). The main tool is a ∂ ‾ -generalization of the Deift-Zhou nonlinear steepest descent method. [ABSTRACT FROM AUTHOR]

In this paper, we define a general analytical domain and two reflection coefficients to study the long-time asymptotics for a defocusing mixed nonlinear Schrödinger equation with the Schwartz initial data. It is shown that the solution of the initial value problem for the mixed Schrödinger equation can be expressed in terms of the solution associated with a Riemann-Hilbert problem. The long-time asymptotic of the mixed Schrödinger equation is further obtained via the Deift-Zhou nonlinear steepest descent method. The long-time asymptotic for the classical Schrödinger equation, derivative Schrödinger equation and modified Schrödinger equation are obtained directly from our results as special cases. [ABSTRACT FROM AUTHOR]