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International Journal of Modern Physics B

THREE-WAVE SOLITON EXCITATIONS IN A DISK-SHAPED BOSE–EINSTEIN CONDENSATE

CHUNLIU SUN, Yousheng Xu, Weina Cui, Guoxiang Huang, Jacob Szeftel, Bambi Hu

Published Sep 20, 2005• 5 citations
Publisher

Abstract

A three-wave resonant interaction of collective modes and related soliton excitations in a disk-shaped Bose–Einstein condensate are investigated. The phase-matching conditions for the resonant interaction are satisfied by suitably choosing the wavevectors and the frequencies of the collective modes. A set of nonlinearly coupled envelope equations describing the spatio-temporal evolution of the three-wave resonant interaction are derived by using a method of multiple-scales, and some explicit (2+1)-dimensional three-wave soliton solutions are also presented and discussed.

DOI
10.1142/s0217979205032322

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A three-wave resonant interaction of collective modes and related soliton excitations in a disk-shaped Bose–Einstein condensate are investigated. The phase-matching conditions for the resonant interaction are satisfied by suitably choosing the wavevectors and the frequencies of the collective modes. A set of nonlinearly coupled envelope equations describing the spatio-temporal evolution of the three-wave resonant interaction are derived by using a method of multiple-scales, and some explicit (2+1)-dimensional three-wave soliton solutions are also presented and discussed.
We investigate the three-wave resonant interaction (TWRI) of Bogoliubov excitations in a disk-shaped Bose–Einstein condensate with the diffraction of the excitations taken into account. We show that the phase-matching condition for the TWRI can be satisfied by a suitable selection of the wavevectors and the frequencies of the three exciting modes involved in the TWRI. Using a method of multiple-scales we derive a set of nonlinearly coupled envelope equations describing the TWRI process and give some explicit solitary-wave solutions.
We make a systematic study of the resonant mode coupling of the collective excitations at zero temperature in a Bose-Einstein condensate (BEC). (i) Based on the Gross-Pitaevskii equation we derive a set of nonlinearly coupled envelope equations for a three-mode resonant interaction (TMRI) by means of a method of multiple scales. (ii) We calculate the coupling matrix elements for the TMRI and show that the divergence appearing in previous studies can be eliminated completely by using a Fetter-like variational approximation for the ground-state wave function of the condensate. (iii) We provide the selection rules in mode-mode interaction processes [including TMRI and second-harmonic generation (SHG)] according to the symmetry of the excitations. (iv) By solving the nonlinearly coupled envelope equations we obtain divergence-free nonlinear amplitudes for the TMRI and SHG processes and show that our theoretical results on the shape oscillations of the condensate agree well with the experimental ones. We suggest also an experiment to check the theoretical prediction of the present study on the TMRI of collective excitations in a BEC.
The resonant interaction between collective modes in a quasi-one-dimensional (Q1D) attractive Bose–Einstein condensate (BEC) is considered. Nonlinearly coupled amplitude equations are derived using a method of multiple scales. Coupling matrix elements describing mode–mode resonances in both Landau and Beliaev mechanisms are calculated analytically. In particular, second-harmonic generation and three-mode resonant interaction of the collective modes in Q1D BEC are investigated in detail. The result shows that the physics of the resonant mode couplings in attractive Q1D BEC is richer than that of repulsive Q1D BEC.