Related papers: Continuum limit of 3D fractional nonlinear Schr\"o…
From the mathematical side, nonlinear Schr\"odinger equations are usually investigated via variational methods, that cease to work in higher dimensions. This thesis tries to overcome this problem by focusing on spherically symmetric…
We consider the nonlinear Schr\"odinger equation with periodic dispersion management. We first establish global-in-time Strichartz estimates for the underlying linear equation with suitable dispersion maps. As an application, we establish a…
The aim of this work is to study the numerical solution of the nonlinear Schrodinger problem using a combination between Witt basis and finite difference approximations. We construct a discrete fundamental solution for the non-stationary…
In this paper, we first investigate the global existence of a solution for the stochastic fractional nonlinear Schr\"odinger equation with radially symmetric initial data in a suitable energy space $H^{\alpha}$. We then show that the…
Integrable fractional equations such as the fractional Korteweg-deVries and nonlinear Schr\"odinger equations are key to the intersection of nonlinear dynamics and fractional calculus. In this manuscript, the first discrete/differential…
We study positive bound states for the semiclassical stationary nonlinear Schr\"odinger equation. We are especially interested in solutions which concentrate on a lower dimensional sphere. We adopt a purely variational approach which allows…
We consider the energy critical nonlinear Schr\"odinger equation on periodic domains of the form R^m x T^{4-m} with m=0,1,2,3. Assuming that a certain L^4 Strichartz estimate holds for solutions to the corresponding linear Schr\"odinger…
We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension, $$i\partial_t u - \Lambda u = c_0{|u|}^2 u + c_1 u^3 + c_2 u \bar{u}^2 + c_3 \bar{u}^3,…
For the 3d cubic nonlinear Schr\"odinger (NLS) equation, which has critical (scaling) norms $L^3$ and $\dot H^{1/2}$, we first prove a result establishing sufficient conditions for global existence and sufficient conditions for finite-time…
We consider the cubic defocusing nonlinear Schr\"odinger equation in one dimension with the nonlinearity concentrated at a single point. We prove global well-posedness in the scaling-critical space $L^2(\mathbb{R})$ and scattering for all…
In this paper we address the question of the pointwise almost everywhere limit of nonlinear Schr\"odinger flows to the initial data, in both the continuous and the periodic settings. Then we show how, in some cases, certain smoothing…
In this article, we aim to study the scattering of the solution to the focusing inhomogeneous nonlinear Schr\"odinger equation with a potential of form \begin{align*} i\partial_t u+\Delta u- Vu=-|x|^{-b}|u|^{p-1}u \end{align*} in the energy…
We consider a discrete nonlinear Schr\"odinger equation with long-range interactions and a memory effect on the infinite lattice $h\Z$ with mesh-size $h>0$. Such models are common in the study of charge and energy transport in biomolecules.…
Scattering for the mass-critical fractional Schr\"odinger equation with a cubic Hartree-type nonlinearity for initial data in a small ball in the scale-invariant space of three-dimensional radial and square-integrable initial data is…
The norm resolvent convergence of discrete Schr\"odinger operators to a continuum Schr\"odinger operator in the continuum limit is proved under relatively weak assumptions. This result implies, in particular, the convergence of the spectrum…
We consider the Cauchy problem to the 3D fractional Schr\"odinger equation with quadratic interaction of $u\bar u$ type. We prove the global existence of solutions and scattering properties for small initial data. For the proof, one novelty…
In this paper, we consider the following three dimensional defocusing cubic nonlinear Schr\"odinger equation (NLS) with partial harmonic potential \begin{equation*}\tag{NLS} i\partial_t u + \left(\Delta_{\mathbb{R}^3 }-x^2 \right) u = |u|^2…
This paper proves endpoint Strichartz estimates for the linear Schroedinger equation in $R^3$, with a time-dependent potential that keeps a constant profile and is subject to a rough motion, which need not be differentiable and may be large…
We show that the thermodynamics of the focusing cubic discrete nonlinear Schrodinger equation are exactly solvable in dimensions three and higher. A number of explicit formulas are derived.
We introduce and analyze a symmetric low-regularity scheme for the nonlinear Schr\"odinger (NLS) equation beyond classical Fourier-based techniques. We show fractional convergence of the scheme in $L^2$-norm, from first up to second order,…