Related papers: Construction of a solution for the two-component r…
We study the formation of singularities for cylindrical symmetric solutions to the Gross-Pitaevskii equation describing a dipolar Bose-Einstein condensate. We prove that solutions arising from initial data with energy below the energy of…
We consider second order elliptic divergence form systems with complex measurable coefficients $A$ that are independent of the transversal coordinate, and prove that the set of $A$ for which the boundary value problem with $L_2$ Dirichlet…
We study minimizers of a Gross--Pitaevskii energy describing a two-component Bose-Einstein condensate confined in a radially symmetric harmonic trap and set into rotation. We consider the case of coexistence of the components in the…
The time dependent Gross-Pitaevskii equation describes the dynamics of initially trapped Bose-Einstein condensates. We present a rigorous proof of this fact starting from a many-body bosonic Schroedinger equation with a short scale…
Within the formalism of the Gross-Pitaevskii equation, we derive effective one- and two-dimensional equations for cigar- and pancake-shaped dipolar Bose-Einstein condensates with arbitrary polarization angle. These are based on an ansatz…
A general stability criterion is derived for the D-dimensional ground states of the Gross-Pitaevskii equation, which describes attractive Bose-Einstein condensates confined in a magnetic trap. These ground states are shown to avoid the…
It was recently shown by the authors that a semilinear elliptic equation can be represented as an infinite-dimensional dynamical system in terms of boundary data on a shrinking one-parameter family of domains. The resulting system is…
We construct a fully self-consistent non-equilibrium theory for the dynamics of two interacting finite-temperature atomic Bose-Einstein condensates. The condensates are described by dissipative Gross-Pitaevskii equations, coupled to quantum…
In the first part of this series, an augmented PDE system was introduced in order to couple two nonlinear hyperbolic equations together. This formulation allowed the authors, based on Dafermos's self-similar viscosity method, to establish…
We analyse a generalised Gross-Pitaevskii equation involving a paraboloidal trap potential in $D$ space dimensions and generalised to a nonlinearity of order $2n+1$. For {\em attractive} coupling constants collapse of the particle density…
The focusing of a propagating untrapped Bose-Einstein condensate is studied theoretically. We use a scaling solution method comprising a time-dependent scaling function to analytically examine the dynamics of a falling Bose-Einstein…
In this paper, we study the coupled Einstein constraint equations on complete manifolds through the conformal method, focusing on non-compact manifolds with flexible asymptotics. This is physically well-motivated by standard cosmological…
Coupled nonlinear Schrodinger equations for paraxial optics with two circular polarizations of light in a defocusing Kerr medium with anomalous dispersion coincide in form with the Gross-Pitaevskii equations for a binary Bose-Einstein…
A Petrov-Galerkin finite element method is constructed for a singularly perturbed elliptic problem in two space dimensions. The solution contains a regular boundary layer and two characteristic boundary layers. Exponential splines are used…
We consider a weakly coupled singularly perturbed variational elliptic system in a bounded smooth domain with Dirichlet boundary conditions. We show that, in the competitive regime, the number of fully nontrivial solutions with nonnegative…
We address a two-dimensional nonlinear elliptic problem with a finite-amplitude periodic potential. For a class of separable symmetric potentials, we study the bifurcation of the first band gap in the spectrum of the linear Schr\"{o}dinger…
By exact numerical solutions of the Gross-Pitaevskii (GP) equation in 3D, we assess the validity of 1D and 2D approximations in the study of Bose-Einstein condensates confined in harmonic trap potentials. Typically, these approximations are…
We consider the mean-field vortex solutions and their stability within a two-component Bose Einstein condensate in the immiscible limit. A variational approach is employed to study a system consisting of a majority component which contains…
We study a double Cahn-Hilliard type functional related to the Gross-Pitaevskii energy of two-components Bose-Einstein condensates. In the case of large but same order intercomponent and intracomponent coupling strengths, we prove…
We study one dimensional mixtures of two-component Bose-Einstein condensates in the limit where the intra-species and inter-species interaction constants are very close. Near the mixing-demixing transition the polarization and the density…