Related papers: The derivative NLS equation: global existence with…
We prove the existence of global solutions to the DNLS equation with initial data in a large subset of $H^2(\mathbb R)\cap H^{1,1}(\mathbb R)$ containing a neighborhood of all solitons. We use the inverse scattering transform method, which…
We address existence of global solutions to the derivative nonlinear Schr\"{o}dinger (DNLS) equation without the small-norm assumption. By using the inverse scattering transform method without eigenvalues and resonances, we construct a…
The global existence of the solution for the second-type derivative nonlinear Schr\"odinger (DNLSII) equation with solitons is presented for the first time on the line with weighted Sobolev initial data in $H^2( \mathbb{R}) \cap…
We address the existence of global solutions to the initial value problem for the integrable nonlocal derivative nonlinear Schr\"{o}dinger equation in weighted Sobolev space $H^{2}(\mathbb{R})\cap H^{1,1}(\mathbb{R})$. The key to prove this…
We review recent results on global wellposedness and long-time behavior of smooth solutions to the derivative nonlinear Schr\"{o}dinger (DNLS) equation. Using the integrable character of DNLS, we show how the inverse scattering tools and…
We give a sufficient condition for global existence of the solutions to a generalized derivative nonlinear Schr\"{o}dinger equation (gDNLS) by a variational argument. The variational argument is applicable to a cubic derivative nonlinear…
We consider the one-dimensional focusing nonlinear Schr\"odinger equation (NLS) with a delta potential and even initial data. The problem is equivalent to the solution of the initial/boundary problem for NLS on a half-line with Robin…
We consider the Cauchy problem for the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation $ \I\partial_t q(x,t)+\partial_{x}^2q(x,t)+2\sigma q^{2}(x,t)\overline{q(-x,t)}=0 $ with initial data $q(x,0)\in H^{1,1}(\mathbb{R})$. It is…
We consider the NLS on Schwarzschild manifold.For radial solutions with sufficiently localized initial data,global existence,L^p estimates and asymptotic completeness of the wave operators is proved
In this article we study the asymptotic behavior of a quadratic NLS equation with small, time-dependent potential and small spatially localized initial data. We prove global existence and scattering of solutions. The two main ingredients of…
In this paper we investigate the global well-posedness and long-term behavior of solutions to the kinetic derivative nonlinear Schr\"odinger equation (KDNLS) on the real line. The equation incorporates both local cubic nonlinearities with…
We develop inverse scattering for the derivative nonlinear Schrodinger equation (DNLS) on the line using its gauge equivalence with a related nonlinear dispersive equation. We prove Lipschitz continuity of the direct and inverse scattering…
We consider the Cauchy problem for the defocusing nonlinear Schr\"odinger equations (NLS) on the real line with a special subclass of almost periodic functions as initial data. In particular, we prove global existence of solutions to NLS…
We consider the inverse scattering transform for the nonlinear Schr\"{o}dinger equation in laboratory coordinates (NLSLab equation) with nonzero boundary conditions (NZBCs) at infinity. In order to better deal with the scattering problem of…
In this paper, we investigate the space-time shifted nonlocal derivative nonlinear Schr\"{o}dinger (DNLS) equation under nonzero boundary conditions using the Riemann--Hilbert (RH) approach for the first time. To begin with, in the direct…
We study the Cauchy problem for the defocusing nonlinear Schr\"odinger (NLS) equation under the assumption that the solution vanishes as $x \to + \infty$ and approaches an oscillatory plane wave as $x \to -\infty$. We first develop an…
We show that the derivative nonlinear Schr\"odinger (DNLS) equation is globally well-posed in the weighted Sobolev space $H^{2,2}(\mathbb{R})$. Our result exploits the complete integrability of DNLS and removes certain spectral conditions…
In this paper, we consider the derivative nonlinear Schr\"odinger (DNLS) equation. While the existence theory has been intensely studied, properties like dispersive estimates for the solutions have not yet been investigated. Here we address…
We study the Derivative Nonlinear Schr\"odinger (DNLS). equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities corresponding to algebraic solitons). We show…
In 2013 a new nonlocal symmetry reduction of the well-known AKNS scattering problem was found; it was shown to give rise to a new nonlocal $PT$ symmetric and integrable Hamiltonian nonlinear Schr\"{o}dinger (NLS) equation. Subsequently, the…