Related papers: The Derivation of the $\mathbb{T}^{3}$ Energy-crit…
We derive the defocusing cubic Gross-Pitaevskii (GP) hierarchy in dimension $d=3$, from an $N$-body Schr\"{o}dinger equation describing a gas of interacting bosons in the GP scaling, in the limit $N\rightarrow\infty$. The main result of…
In this paper we study the derivation of a certain type of NLS from many-body interactions of bosonic particles. We consider a model with a finite linear combination of $n$-body interactions, where $n \geq 2$ is an integer. We show that the…
We investigate the dynamics of a boson gas with three-body interactions in dimensions $d=1,2$. We prove that in the limit of infinite particle number, the BBGKY hierarchy of $k$-particle marginals converges to a limiting (Gross-Pitaevskii…
In this paper, we study solutions to the three-dimensional quintic Gross-Pitaevskii hierarchy. We prove unconditional uniqueness among all small solutions in the critical space $\mathfrak{H}^1$ (which corresponds to $H^1$ on the NLS level).…
We consider a 2D time-dependent quantum system of $N$-bosons with harmonic external confining and \emph{attractive} interparticle interaction in the Gross-Pitaevskii scaling. We derive stability of matter type estimates showing that the…
We consider the dynamics of $N$ bosons in three dimensions. We assume the pair interaction is given by $N^{3\beta -1}V(N^{\beta }\cdot )$ . By studying an associated many-body wave operator, we introduce a BBGKY hierarchy which takes into…
We derive the 3D quintic NLS as the mean field limit of a Bose gas with three-body interactions. The quintic NLS is energy-critical, leading to several new difficulties in comparison with the cubic NLS which emerges from Bose gases with…
We consider the $\mathbb{T}^{4}$ cubic NLS which is energy-critical. We study the unconditional uniqueness of solution to the NLS via the cubic Gross-Pitaevskii hierarchy, an uncommon method, and does not require the existence of solution…
We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is attractive and given by $a^{3\beta-1}V(a^{\beta}\cdot)$…
We consider the 3D quantum many-body dynamics describing a dilute bose gas with strong confining in one direction. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential…
We derive rigorously, for both R^2 and [-L, L]^2, the cubic nonlinear Schrodinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence…
We consider the dynamical Gross-Pitaevskii (GP) hierarchy on $\R^d$, $d\geq1$, for cubic, quintic, focusing and defocusing interactions. For both the focusing and defocusing case, and any $d\geq1$, we prove local existence and uniqueness 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 the (de)focusing cubic Gross-Pitaevskii (GP) hierarchy on $\mathbb{R}$, which is an infinite hierarchy of coupled linear inhomogeneous PDE which appears in the derivation of the cubic nonlinear Schr\"{o}dinger (NLS) equation…
In this work, we obtain a microscopic derivation of Gibbs measures for the focusing quintic nonlinear Schr\"{o}dinger equation (NLS) on $\mathbb{T}$ from many-body quantum Gibbs states. On the quantum many-body level, the quintic…
We construct solutions with prescribed scattering state to the cubic-quintic NLS $$ (i\partial_t+\Delta)\psi=\alpha_1 \psi-\alpha_{3}\vert \psi\vert^2 \psi+\alpha_5\vert \psi\vert^4 \psi $$ in three spatial dimensions in the class of…
We study the growth of correlations in systems with weak long-range interactions. Starting from the BBGKY hierarchy, we determine the evolution of the two-body correlation function by using an expansion of the solutions of the hierarchy in…
We extend our previous result on the nonlinear Klein-Gordon equation to the nonlinear Schrodinger equation with the focusing cubic nonlinearity in three dimensions, for radial data of energy at most slightly above that of the ground state.…
Consider the defocusing quintic nonlinear Schr\"{o}dinger equation on $\mathbf{R}^3$ with initial data in the energy space. This problem is "energy-critical" in view of a certain scale-invariance, which is a main source of difficulty in the…
We consider the derivation of the defocusing cubic nonlinear Schr\"{o}dinger equation (NLS) on $\mathbb{R}^{3}$ from quantum $N$-body dynamics. We reformat the hierarchy approach with Klainerman-Machedon theory and prove a bi-scattering…