Dynamics of Bose-Einstein Condensates: Exact Representation and Topological Classification of Coherent Matter Waves

Leilei Jia,Qihuai Liu,Shengqiang Tang

doi:10.1155/2014/156513

Abstract

By using the bifurcation theory of dynamical systems, we present the exact representation and topological classification of coherent matter waves in Bose-Einstein condensates (BECs), such as solitary waves and modulate amplitude waves (MAWs). The existence and multiplicity of such waves are determined by the parameter regions selected. The results show that the characteristic of coherent matter waves can be determined by the “angular momentum” in attractive BECs while for repulsive BECs; the waves of the coherent form are all MAWs. All exact explicit parametric representations of the above waves are exhibited and numerical simulations support the result.

Highlights

Particles in a dilute gas reside in the same quantum state at low temperatures, which form a Bose-Einstein condensate (BEC)
The spatial dynamics of amplitudes R(x) for the wave functions in BECs are determined by nonlinear system (7), which depends on the parameter variables c, g, δ
Let us fix the parameters g = −1, δ = 50 which are corresponding to the attractive interactions in BECs

Summary

Introduction

Particles in a dilute gas reside in the same quantum (ground) state at low temperatures, which form a Bose-Einstein condensate (BEC). This phenomenon was first observed experimentally in 1995 with vapors of rubidium and sodium [1]. While the small perturbed parameter ε ≠ 0, for example, Bose-Einstein condensed atoms being perturbed by a weak optical lattice potential, the dynamical characters of these solutions may be preserved [14, 15] as periodic or quasiperiodic modulated amplitude waves. We will give a full topological classification of coherent structural solutions for system (1) with various adjustable parameters, while restricting the external potentials V(x) ≡ V0. This phenomenon has been studied in other physical models by some authors (see [20, 21])

Coherent Structures

Bifurcations of Phase Portraits

Exact Explicit Representations of Matter Waves

Numerical Simulations