Related papers: Derivation of the two dimensional nonlinear Schrod…
We derive the cubic defocusing GP hierarchy in ${\mathbb R}^3$ from a bosonic $N$-particle Schr\"odinger equation as $N\rightarrow\infty$, in the strong topology corresponding to the space ${\mathcal H}_\xi^1$ introduced in \cite{chpa}. In…
Consider the following system of double coupled Schr\"odinger equations arising from Bose-Einstein condensates etc., \begin{equation*} \left\{\begin{array}{l} -\Delta u + u =\mu_1 u^3 + \beta uv^2- \kappa v, -\Delta v + v =\mu_2 v^3 + \beta…
We consider the dynamics of a 2D Bose gas with an interaction potential of the form $N^{2\beta-1}w(N^\beta\cdot)$ for $\beta\in (0,3/2)$. The interaction may be chosen to be negative and large, leading to the instability regime where the…
We consider the dynamics of $N$ interacting bosons initially exhibiting Bose-Einstein condensation. Due to an external trapping potential, the bosons are strongly confined in two spatial directions, with the transverse extension of the trap…
We propose a Lie-algebraic duality approach to analyze non-equilibrium evolution of closed dynamical systems and thermodynamics of interacting quantum lattice models (formulated in terms of Hubbard-Stratonovich dynamical systems). The first…
We consider the 2d and 3d many body Schr\"odinger equations in the presence of anisotropic switchable quadratic traps. We extend and improve the collapsing estimates in Klainerman-Machedon [24] and Kirkpatrick-Schlein-Staffilani [23].…
Starting from first principle many-body quantum dynamics, we show that the dynamics of Bose-Einstein condensates can be approximated by the time-dependent nonlinear Gross-Pitaevskii equation, giving a bound on the rate of the convergence.…
A mathematical framework for optimal bilinear control of nonlinear Schr\"odinger equations of Gross-Pitaevskii type arising in the description of Bose-Einstein condensates is presented. The obtained results generalize earlier efforts found…
In this note, we generalize the nonlinearity-recovery result in [7] for classical cubic nonlinear Schr\"odinger equations to higher-order Schr\"odinger equations with a more general nonlinearity. More precisely, we consider a…
We provide a rigorous derivation of nonlinear Gibbs measures in two and three space dimensions, starting from many-body quantum systems in thermal equilibrium. More precisely, we prove that the grand-canonical Gibbs state of a large bosonic…
For the first time, a nonlinear Schr\"odinger equation of the general form is considered, depending on time and two spatial variables, the potential and dispersion of which are specified by two arbitrary functions. This equation naturally…
We study a first-order hyperbolic approximation of the nonlinear Schr\"odinger (NLS) equation. We show that the system is strictly hyperbolic and possesses a modified Hamiltonian structure, along with at least three conserved quantities…
We consider the dynamics of $N$ interacting bosons initially forming a Bose-Einstein condensate. Due to an external trapping potential, the bosons are strongly confined in two dimensions, where the transverse extension of the trap is of…
The cubic nonlinear Schr\"odinger equation with repulsive nonlinearity and an elliptic function potential models a quasi-one-dimensional repulsive dilute gas Bose-Einstein condensate trapped in a standing light wave. New families of…
We derive rigorously the 2D periodic focusing cubic NLS as the mean-field limit of the 3D focusing quantum many-body dynamics describing a dilute Bose gas with periodic boundary condition in the $x$-direction and a well of infinite-depth in…
We consider multiple solutions to the nonlinear Schr\"odinger equation (NLS) with a partial confinement, which is physically relevant to dynamics of the Bose-Einstein condensate. Our study not only verifies the existence of positive ground…
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…
This article consists of two parts. In the first part, we review the most recent proofs establishing quadratic Morawetz inequalities for the nonlinear Schr\"odinger equation (NLS). We also describe the applications of these estimates to the…
For the numerical solution of the cubic nonlinear Schr\"{o}dinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to…
For nonlinear dispersive systems, the nonlinear Schr\"odinger (NLS) equation can usually be derived as a formal approximation equation describing slow spatial and temporal modulations of the envelope of a spatially and temporally…