Related papers: Dispersive estimates and NLS on product manifolds

We obtain polynomial bounds on the growth in time of Sobolev norm of solutions to the cubic defocusing nonlinear Schrodinger equation on two dimensional product space. We also give the angular improved bilinear Strichartz estimates for…

Schr\"odinger equations with nonlinearities concentrated in some regions of space are good models of various physical situations and have interesting mathematical properties. We show that in the semiclassical limit it is possible to…

Let $(M,g)$ be a compact smooth $3$-dimensional Riemannian manifold without boundary. It is proved that the energy-critical nonlinear Schr\"odinger equation is globally well-posed for small initial data in $H^1(M)$, provided that a certain…

We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Schr\"odinger and wave equations. In particular, we improve upon previous works and weaken the conditions on the potentials. To this end we also provide…

We study a quite general class of stochastic dispersive equations with linear multiplicative noise, including especially the Schr\"odinger and Airy equations. The pathwise Strichartz and local smoothing estimates are derived here in both…

We prove dispersive and Strichartz estimates for Schr\"o- dinger equations on a class of locally symmetric spaces {\Gamma}\X, where X = G/K is a symmetric space and {\Gamma} is a torsion free discrete sub- group of G. We deal with the cases…

We extend a result on dispersion for solutions of the linear Schr\"odinger equation, proved by Firsova for operators with finitely many energy bands only, to the case of smooth potentials in 1D with infinitely many bands. The proof consists…

We study the stochastic nonlinear Schr\"odinger equations with additive stochastic forcing. By using the dispersive estimate, we present a simple argument, constructing a unique local-in-time solution with rougher stochastic forcing than…

We establish Strichartz estimates for the Schr\"odinger equation on Riemannian manifolds $(\Omega,\g)$ with boundary, for both the compact case and the case that $\Omega$ is the exterior of a smooth, non-trapping obstacle in Euclidean…

In this article, we prove a bilinear estimate for Schr\"odinger equations on 2d waveguide, $\mathbb{R}\times \mathbb{T}$. We hope it may be of use in the further study of concentration compactness for cubic NLS on $\mathbb{R}\times…

We prove a sharp dispersive estimate $$ |P_{ac}u(t,x)|\le C|t|^{-1/2}\|u(0)\|_{L^1(R)} $$ for the one dimensional Schr\"odinger equation $$ iu_{t}-u_{xx}+V(x)u+V_0 u=0, $$ where $(1+x^2)V\in L^1(R)$ and $V_0$ is a step function, real valued…

We consider the Cauchy problem for dispersion managed nonlinear Schroedinger equations, where the dispersion map is assumed to be periodic and piecewise constant in time. We establish local and global well-posedness results and the…

We prove Strichartz estimates with a loss of derivatives for the Schr\"odinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates the on polygon…

We prove dispersive estimates for solutions to the Schrodinger equation with a real-valued potential $V\in L^\infty({\bf R}^n)$, $n\ge 4$, satisfying $V(x)=O(|x|^{-(n+2)/2-\epsilon})$, $|x|>1$, $\epsilon>0$.

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

The interface problem for the linear Schr\"odinger equation in one-dimensional piecewise homogeneous domains is examined by providing an explicit solution in each domain. The location of the interfaces is known and the continuity of the…

In this paper we present a method to study global regularity properties of solutions of large-data critical Schrodinger equations on certain noncompact Riemannian manifolds. We rely on concentration compactness arguments and a global…

We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential…

A one-dimensional generalized nonlinear Schroedinger equation is considered, and the corresponding inverse scattering problem is analyzed when the potential is compactly supported and depends on the wave function. The unique recovery of the…

We prove dispersive estimates for two models~: the adjacency matrix on a discrete regular tree, and the Schr\"odinger equation on a metric regular tree with the same potential on each edge/vertex. The latter model can be thought of as an…