In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

In this paper, we mainly review recent results on mathematical theory and
numerical methods for Bose-Einstein condensation (BEC),
based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are then
carried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models for
spin-1 BEC, Bogoliubov excitation and BEC at finite temperature.

http://osada.csp.escience.cn/dct/attach/Y2xiOmNsYjpwZGY6NjI5NzY=

Mathematical theory and numerical methods for Bose-Einstein condensation

In this paper, we mainly review recent results on mathematical theory and numerical methods for Bose-Einstein condensation (BEC), based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are then carried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models for spin-1 BEC, Bogoliubov excitation and BEC at finite temperature.

In this paper, we mainly review recent results on mathematical theory and numerical methods for Bose-Einstein condensation (BEC), based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduc- tion of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the nite time blow-up. To compute the ground state, the gradient ow with discrete normalization (or imaginary time) method is reviewed and vari- ous full discretization methods are presented and compared. To simulate the dynamics, both nite dierence methods and time splitting spectral methods are reviewed, and their error estimates are briey outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime ( Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are then carried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briey the mathematical models for spin-1 BEC, Bogoliubov excitation and BEC at nite temperature.

Mathematical theory and numerical methods for

We analyze finite difference methods for the Gross-Pitaevskii equation with an angular momentum rotation term in two and three dimensions and obtain the optimal convergence rate, for the conservative Crank-Nicolson finite difference (CNFD) method and semi-implicit finite difference (SIFD) method, at the order of O(h2 + τ2) in the l2-norm and discrete H1-norm with time step τ and mesh size h. Besides the standard techniques of the energy method, the key technique in the analysis for the SIFD method is to use the mathematical induction, and resp., for the CNFD method is to obtain a priori bound of the numerical solution in the l∞-norm by using the inverse inequality and the l2-norm error estimate. In addition, for the SIFD method, we also derive error bounds on the errors between the mass and energy in the discretized level and their corresponding continuous counterparts, respectively, which are at the same order of the convergence rate as that of the numerical solution itself. Finally, numerical results are reported to confirm our error estimates of the numerical methods.

/pdf/optimal-error-estimates-of-finite-difference-methods-for-the-4qgd3aog2u.pdf

Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation

http://www.math.ac.cn/kyry/hcc/pub/papers/201409/P020150618516553813608.pdf

Wellposedness for the fourth order nonlinear Schrödinger equations

http://www.math.ac.cn/kyry/hcc/pub/papers/201409/P020150618516142899495.pdf

Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces

In this paper, we establish the global well posedness of the Cauchy problem for the Gross-Pitaevskii equation with an angular momentum rotational term in which the angular velocity is equal to the isotropic trapping frequency in the space R3.

/pdf/global-well-posedness-for-the-gross-pitaevskii-equation-with-okjqobemq4.pdf

Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions

In this paper, we establish the global well posedness of the Cauchy problem for the Gross–Pitaevskii equation with a rotational angular momentum term in the space ℝ2. Copyright © 2007 John Wiley & Sons, Ltd.

/pdf/global-well-posedness-for-the-gross-pitaevskii-equation-with-4ionyg6jnh.pdf

Global well posedness for the Gross–Pitaevskii equation with an angular momentum rotational term

In this paper, we establish the global well-posedness of the Cauchy problem for the Gross-Pitaevskii equation with an rotational angular momentum term in the space $\Real^2$.

Ling Hsiao

Papers

Wellposedness for the fourth order nonlinear Schrödinger equations

Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces

Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions

Global well posedness for the Gross–Pitaevskii equation with an angular momentum rotational term

Global well-posedness for the Gross-Pitaevskii equation with an angular momentum rotational term