Related papers: Global Smooth Solution of Nonlinear Schr\"odinger …

The Cauchy problem for the derivative nonlinear Schr\"odinger equation with periodic boundary condition is considered. Local well-posedness for periodic initial data u_0 in the space ^H^s_r, defined by the norms ||u_0||_{^H^s_r}=||<xi>^s…

We consider the Cauchy problem for nonlinear Schrodinger equations in the presence of a smooth, possibly unbounded, potential. No assumption is made on the sign of the potential. If the potential grows at most linearly at infinity, we…

We study the solution theory of the nonlinear Schr\"odinger equation with a concentrated nonlinearity on the torus. In particular, we establish existence and uniqueness of global energy-conserving solutions for initial data in $H^1$. Our…

This paper addresses the Cauchy problem for the cubic defocusing nonlinear Schr\"odinger equation (NLS) with almost periodic initial data. We prove that for small analytic quasiperiodic initial data satisfying Diophantine frequency…

We prove the global existence of solution to the small data mass critical stochastic nonlinear Schr\"{o}dinger equation in $d=1$. We further show the stability of the solution under perturbation of initial data. Our construction starts with…

We prove two new results about the Cauchy problem for nonlinear Schroedinger equations on four-dimensional compact manifolds. The first one concerns global wellposedness for Hartree-type nonlinearities and includes approximations of cubic…

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schr\"odinger equation with initial data $u_{0}\in X$, where $X\in\{M_{2,q}^{s}(\mathbb R), H^{\sigma}(\mathbb T),…

In this work, we mainly focus on the energy-supercritical nonlinear Schr\"odinger equation, $$ i\partial_{t}u+\Delta u= \mu|u|^p u, \quad (t,x)\in \mathbb{R}^{d+1}, $$ with $\mu=\pm1$ and $p>\frac4{d-2}$. %In this work, we consider the…

We consider the Cauchy problem for the nonlinear Schr\"odinger equation on $\mathbb{R}^2$, $iu_t + u_{xx} + u_{yy} + \lambda|u|^\sigma u =0$, $\lambda\in \mathbb{R}$, $\sigma>0$. We introduce new functional spaces over which the initial…

Let (M,g) be a three-dimensional smooth compact Riemannian manifold such that all geodesics are simple and closed with a common minimal period, such as the 3-sphere S^3 with canonical metric. In this work the global well-posedness problem…

By introducing a class of new function spaces $B^{\sigma,s}_{p,q}$ as the resolution spaces, we study the Cauchy problem for the nonlinear Klein-Gordon equation (NLKG) in all spatial dimensions $d \geqslant 1$, $$ \partial^2_t u + u- \Delta…

The results of this paper are twofold. One is that we show the local existence and uniqueness of very regular or smooth solution to the initial-Neumann boundary value problem of the Schr\"{o}dinger flow for maps from a smooth bounded domain…

In this paper, we consider the nonlinear Schr\"odinger equation, $$ i\partial_{t}u+\Delta u= \mu|u|^p u, \quad (t,x)\in \mathbb{R}^{d+1}, $$ with $\mu=\pm1, p>0$. In this work, we consider the mass-subcritical cases, that is, $p\in…

We study the Cauchy problem for the non-linear Schr\"odinger equation with singular potentials. For point-mass potential and nonperiodic case, we prove existence and asymptotic stability of global solutions in weak-L^{p} spaces. Specific…

We study the nonlinear Schr\"odinger equation (NLS) with bounded initial data which does not vanish at infinity. Examples include periodic, quasi-periodic and random initial data. On the lattice we prove that solutions are polynomially…

We study the Cauchy problem for the cubic nonlinear Schroedinger equation, perturbed by (higher order) dissipative nonlinearities. We prove global in-time existence of solutions for general initial data in the energy space. In particular we…

In this paper, we consider the defocusing nonlinear Schr\"odinger equation in space dimensions $d\geq 4$. We prove that if $u$ is a radial solution which is \emph{priori} bounded in the critical Sobolev space, that is, $u\in L_t^\infty…

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…

This paper deals with the 2-D Schr\"odinger equation with time-oscillating exponential nonlinearity $i\partial_t u+\Delta u= \theta(\omega t)\big(e^{4\pi|u|^2}-1\big)$, where $\theta$ is a periodic $C^1$-function. We prove that for a class…

We consider the Cauchy problem for the fourth order nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ^2) u= \pm \partial (|u|^2u)$ on $\mathbb{R} ^d$, $d \ge 3$, with random initial data, where…