A multigrid method for the ground state solution of Bose–Einstein condensates based on Newton iteration

BIT Numerical Mathematics - In this paper, a new kind of multigrid method is proposed for the ground state solution of Bose–Einstein condensates based on Newton iteration scheme. Instead of...     

Rotating multicomponent Bose–Einstein condensates

In the understanding of the spatial behavior of interacting components of multicomponent Bose–Einstein condensates (BECs), a central problem is to establish whether coexistence of all the components occurs, or the interspecies interaction leads to extinction, that is, configurations where one or more densities are null. In this paper, for the rotating k-mixture BEC, we prove that the interspecies interaction leads to extinction in the Thomas–Fermi approximation.     

A good approximation of modulated amplitude waves in Bose–Einstein condensates

In this paper, we present a perturbation method that utilizes Hamiltonian perturbation theory and averaging to analyze spatio-temporal structures in Gross–Pitaevskii equations and thereby investigate the dynamics of modulated amplitude waves (MAWs) in quasi-one-dimensional Bose–Einstein condensates with mean-field interactions. A good approximation for MAWs is obtained. We also explore dynamics of BECs with the nonresonant external potentials and scatter lengths varying periodically in detail using Hamiltonian perturbation theory and numerical simulations.     

Orbital stability of standing waves of two-component Bose–Einstein condensates with internal atomic Josephson junction

In this paper, we prove existence, symmetry and uniqueness of standing waves for a coupled Gross–Pitaevskii equations modeling component Bose–Einstein condensates BEC with an internal atomic Josephson junction. We will then address the orbital stability of these standing waves and characterize their orbit.     

Segregation and symmetry breaking of strongly coupled two-component Bose–Einstein condensates in a harmonic trap

We study ground states of two-component condensates in a harmonic trap. We prove that in the strongly coupled and weakly interacting regime, the two components segregate while a symmetry breaking occurs. More precisely, we show that when the intercomponent coupling strength is very large and both intracomponent coupling strengths are small, each component is close to the positive or the negative part of a second eigenfunction of the harmonic oscillator in ${\mathbb{R}^2}$ . As a result, the supports of the components approach complementary half-spaces, and they are not radially symmetric.     

A Cox Process Involved in the Bose–Einstein Condensation

The point process corresponding to the configurations of bosons in standard conditions is a Cox process driven by the square norm of a centered Gaussian process. This point process is infinitely divisible. We point out the fact that this property is preserved by the Bose–Einstein condensation phenomenon and show that the obtained point process after such a condensation occured, is still a Cox process but driven by the square norm of a shifted Gaussian process, the shift depending on the density of the particles. This law provides an illustration of a “super”- Isomorphism Theorem existing above the usual Isomorphism Theorem of Dynkin available for Gaussian processes. Submitted: February 8, 2008. Accepted: March 5, 2008.     

Homotopy–perturbation method for pure nonlinear differential equation

In this paper, the homotopy–perturbation method proposed by J.-H. He is adopted for solving pure strong nonlinear second-order differential equation. For the oscillatory differential equation the initial approximate solution is assumed in the form of Jacobi elliptic function and the forementioned method is used for obtaining of the approximate analytic solution. Two types of differential equations are considered: with strong cubic and strong quadratic nonlinearity. The obtained solution is compared with exact numerical one. The difference between these solutions is negligible for a long time period. The method is found to work extremely well in the examples, but the theoretical reasons are not yet clear.     

Bose–Einstein condensation in satisfiability problems

This paper is concerned with the complex behavior arising in satisfiability problems. We present a new statistical physics-based characterization of the satisfiability problem. Specifically, we design an algorithm that is able to produce graphs starting from a k-SAT instance, in order to analyze them and show whether a Bose–Einstein condensation occurs. We observe that, analogously to complex networks, the networks of k-SAT instances follow Bose statistics and can undergo Bose–Einstein condensation. In particular, k-SAT instances move from a fit-get-rich network to a winner-takes-all network as the ratio of clauses to variables decreases, and the phase transition of k-SAT approximates the critical temperature for the Bose–Einstein condensation. Finally, we employ the fitness-based classification to enhance SAT solvers (e.g., ChainSAT) and obtain the consistently highest performing SAT solver for CNF formulas, and therefore a new class of efficient hardware and software verification tools.     

Bose–Einstein Condensates with Non-classical Vortex

Coupled systems of nonlinear Schrödinger equations have been used extensively to describe Bose–Einstein condensates. In this paper, we study a two-component Bose–Einstein condensate (BEC) with an external driving field in a three-dimensional space. This model gives rise to a new kind of vortex–filaments, with fractional degree and nontrivial core structure. We show that vortex–filaments is 1-rectifiable set, and calculate its mean curvature in the strong coupling (Thomas–Fermi) limit. In particular, we show that large strength of the external driving field causes vortex–filaments for a two-component BEC.     

Applications of some transformations for several variable-coefficient nonlinear evolution equations from plasma physics,arterial mechanics,nonlinear optics and Bose–Einstein condensates

Variable-coefficient nonlinear evolution equations have occurred in such fields as plasma physics, arterial mechanics, nonlinear optics and Bose–Einstein condensates. This paper is devoted to giving some transformations to convert the original nonlinear evolution equations, e.g., the variable-coefficient nonlinear Schrödinger, generalized Gardner and variable-coefficient Sawada–Kotera equations to simpler ones or even constant-coefficient ones. Based on some constraints, we simplify the original equations and derive the associated chirp solitons, Lax pairs, and Bäcklund transformations from the original equations by means of the aforementioned transformations.     

A study of the convergence of and error estimation for the homotopy perturbation method for the Volterra–Fredholm integral equations

In this work the solution of the Volterra–Fredholm integral equations of the second kind is presented. The proposed method is based on the homotopy perturbation method, which consists in constructing the series whose sum is the solution of the problem considered. The problem of the convergence of the series constructed is formulated and a proof of the formulation is given in the work. Additionally, the estimation of the approximate solution errors obtained by taking the partial sums of the series is elaborated on.     

On the rate of convergence to the Bose–Einstein distribution

For a system of identical Bose particles sitting at integer energy levels with the probabilities of microstates given by a multiplicative measure with ≥ 2 degrees of freedom, we estimate the probability of the sequence of occupation numbers to be close to the Bose–Einstein distribution as the total energy tends to infinity. We show that a convergence result earlier proved by A.M. Vershik [Functional Anal. Appl. 30 (2), 95–105 (1996)] is a corollary of our theorems.     

Homotopy perturbation method for coupled Schrödinger–KdV equation

In this paper, numerical analysis of the coupled Schrödinger–KdV equation is studied by using the Homotopy Perturbation Method (HPM). The available analytical solutions of the coupled Schrödinger–KdV equation obtained by multiple traveling wave method are compared with HPM to examine the accuracy of the method. The numerical results validate the convergence and accuracy of the Homotopy Perturbation Method for the analyzed coupled Schrödinger–KdV equation.     

On ground state of spinor Bose�CEinstein condensates

We prove the existence of the ground state for the spinor Bose–Einstein condensates in the one-dimensional case.     

Bounds of the repeated limit for the Bose–Einstein distribution and the construction of partition theory of integers

Mathematical Notes -     

Dynamics of dark–bright vector solitons in a two-component Bose–Einstein condensate

Coupled dark–bright vector solitons are considered in a two-component Bose–Einstein condensate, and their dynamics are investigated by the variational approach based on the renormalized integrals of motion. The stationary states and their atom population distribution are obtained, and it is found that the dark soliton has obvious robust features. The dynamic mechanism is demonstrated by performing a coordinate of a classical particle moving in an effective potential field, and the switching and self-trapping dynamics of the coupled dark–bright vector solitons are discussed by the evolution of the atom population transferring ratio.