Computing Ground States of Bose--Einstein Condensates with Higher Order Interaction via a Regularized Density Function Formulation

Abstract

We propose and analyze a new numerical method for computing the ground state of the modified Gross-Pitaevskii equation for modeling the Bose-Einstein condensate with a higher order interaction by adapting the density function formulation and the accelerated projected gradient method. By reformulating the energy functional E(φ) with φ, the wave function, in terms of the density ρ = |φ| 2 , the original non-convex minimization problem for defining the ground state is then reformulated to a convex minimization problem. In order to overcome the semi-smoothness of the function √ ρ in the kinetic energy part, a regularization is introduced with a small parameter 0 < ε 1. Convergence of the regularization is established when ε → 0. The regularized convex optimization problem is discretized by the second order finite difference method. The convergence rates in terms of the density and energy of the discretization are established. The accelerated projected gradient method is adapted for solving the discretized optimization problem. Numerical results are reported to demonstrate the efficiency and accuracy of the proposed numerical method. Our results show that the proposed method is much more efficient than the existing methods in the literature, especially in the strong interaction regime.

Error estimates of local energy regularization for the logarithmic Schrodinger equation

“…The above regularization saturates the nonlinearity in the region {ρ < ε 2 } (where ρ = |u ε | 2 ), but of course also has some (smaller) effect in the other region {ρ > ε 2 }. Energy regularization has been widely adapted in different areas for dealing with singularity and/or roughness, such as in materials science for establishing the wellposedness of the Cauchy problem for the CH equation with a logarithmic potential [27] and for treating strongly anisotropic surface energy [7,40], in mathematical physics for the well-posedness of the LogSE [17], and in scientific computing on designing regularized numerical methods for treating singularities [9,23,50]. The main aim of this paper is to present an energy regularization for the LogSE (1.1) with three key features as: (i) to first regularize F (ρ) ∈ C 0 ([0, +∞)) in (1.5) (and thus in the energy functional (1.4)), (ii) to regularize the nonlinearity F (ρ) only in the region {ρ < ε 2 } by a sequence of polynomials and keep it unchanged in {ρ > ε 2 } such that the regularized function has given regularity; and (iii) to obtain a sequence of energy regularized logarithmic Schrödinger equations (ERLogSEs) from the regularized energy functional via energy variation.…”