Related papers: Derivation of the two dimensional nonlinear Schrod…
This paper addresses the focusing cubic-quintic nonlinear Schrodinger equation in three space dimensions. Especially, we study the global dynamics of solutions whose energy and mass equal to those of the ground state in the sprits of…
We consider the time evolution of N bosonic particles interacting via a mean field Coulomb potential. Suppose the initial state is a product wavefunction. We show that at any finite time the correlation functions factorize in the limit $N…
We prove that Gibbs measures of nonlinear Schr\"odinger equations arise as high-temperature limits of thermal states in many-body quantum mechanics. Our results hold for defocusing interactions in dimensions $d =1,2,3$. The many-body…
The nonlinear, cubic Schrodinger (NLS) equation has numerous physical applications, but in general is very difficult to solve. Nonetheless, under certain circumstances parameters quantifying the width, momentum and energy of the…
We consider a dispersive equation of Schr{\"o}dinger type with a non-linearity slightly larger than cubic by a logarithmic factor. This equation is supposed to be an effective model for stable two dimensional quantum droplets with LHY…
In this article, we study the two dimensional focusing finitely and infinitely coupled cubic nonlinear Schr\"odinger system when the mass is equal to the scattering threshold. For the focusing finitely coupled cubic nonlinear Schr\"odinger…
In this paper we consider a one-dimensional non-linear Schroedinger equation (NLSE) with a periodic potential. In the semiclassical limit we prove that the stationary solutions of the Bose-Hubbard equation approximate the stationary…
We derive the time-dependent two-component Gross--Pitaevskii (GP) equation as an effective description of the dynamics of a dilute two-component Bose gas near its ground state, which exhibits a two-component Bose-Einstein condensate, in the…
In this paper, we propose a new approach to the relativistic quantum mechanics for many-body, which is a self-consistent system constructed by juxtaposed but mutually coupled nonlinear Dirac's equations. The classical approximation of this…
We consider the quantum many-body dynamics at the weak-coupling scaling. We derive rigorously the quantum Boltzmann equation, which contains the classical hard sphere model and, effectively, the inverse power law model, from the many-body…
In this study, we consider the nonlinear Sch\"odinger equation (NLS) with the zero-boundary condition on a two- or three-dimensional large finite cubic lattice. We prove that its solution converges to that of the NLS on the entire Euclidean…
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…
In this paper a nonlinear coupled Schrodinger system in the presence of mixed cubic and superlinear power laws is considered. A non standard numerical method is developed to approximate the solutions in higher dimensional case. The idea…
We derive an effective equation for the dynamics of many identical bosons in dimension one in the presence of a tiny impurity. The interaction between every pair of bosons is mediated by the impurity through a positive three-body potential.…
We consider NLS on $\T^2$ with multiplicative spatial white noise and nonlinearity between cubic and quartic. We prove global existence, uniqueness and convergence almost surely of solutions to a family of properly regularized and…
The exact solution of the Schr\"odinger equation for the one-dimensional system of interacting particles with the linear dispersion law in an arbitrary external field is found. The solution is reduced to two groups of particles moving with…
We prove non-existence of solutions for the cubic nonlinear Schr\"odinger equation (NLS) on the circle if initial data belong to $H^s(\mathbb{T}) \setminus L^2(\mathbb{T})$ for some $s \in (-\frac18, 0)$. The proof is based on establishing…
The two-dimensional cubic nonlinear Schrodinger equation (NLS) can be used as a model of phenomena in physical systems ranging from waves on deep water to pulses in optical fibers. In this paper, we establish that every one-dimensional…
We consider the final-data problem for systems of nonlinear Schr\"odinger equations with $L^2$ subcritical nonlinearity. An asymptotically free solution is uniquely obtained for almost every randomized asymptotic profile in…
The cubic nonlinear Schrodinger equation with a lattice potential is used to model a periodic dilute gas Bose-Einstein condensate. Both two- and three-dimensional condensates are considered, for atomic species with either repulsive or…