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We obtain the the existence of global solutions to the Cauchy problem of the Fokas-Lenells (FL) equation on the line \begin{align} &u_{xt}+\alpha\beta^2u-2i\alpha\beta u_x-\alpha u_{xx}-i\alpha\beta^2|u|^2u_x=0,\nonumber \\ &u(x,t=0)=u_0(x), \nonumber \end{align} where without the small-norm assumption on initial data $u_0(x)\in H^3(\mathbb{R})\cap H^{2,1}(\mathbb{R})$. 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. The spectral problem associated with the FL equation is changed into an equivalent Zakharov-Shabat-type spectral problem to establish the RH problems on the real axis. 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 priori estimate of the solution to the FL equation. Based on the local solution and the uniformly priori estimate, we construct a unique global solution in $H^3(\mathbb{R})\cap H^{2,1}(\mathbb{R})$ to the FL equation., Comment: 67 pages

We apply $\bar{\partial}$ steepest descent method to obtain sharp asymptotics for a mixed schr��dinger equation $$ q_t+iq_{xx}-ia (\vert q \vert^2q)_x -2b^2\vert q \vert^2q=0,$$ $$q(x,t=0)=q_0(x),$$ under essentially minimal regularity assumptions on initial data in a weighted Sobolev space $q_0(x) \in H^{2,2}(\mathbb{R})$. In the asymptotic expression, the leading order term $\mathcal{O}(t^{-1/2})$ comes from dispersive part $q_t+iq_{xx}$ and the error order $\mathcal{O}(t^{-3/4})$ from a $\overline\partial$ equation, 35 pages

In this paper, we apply $\overline\partial$ steepest descent method to study the Cauchy problem for the derivative nonlinear Schr��dinger equation with nonzero boundary conditions \begin{align} &iq_{t}+q_{xx}+i��(|q|^2q)_{x}=0,\\ & (x,0) = q_0(x), \quad\lim_{x\to\pm\infty} q_0(x) = q_\pm,\end{align} where $|q_\pm|=1$. Based on the spectral analysis of the Lax pair, we express the solution of the derivative nonlinear Schr��dinger equation in terms of solutions of a Riemann-Hilbert problem.In a fixed space-time solitonic region $-3, 47 pages. arXiv admin note: text overlap with arXiv:2005.12208