Nemucial Research On Soliton Solutions Of 1d Gross-Pitaevskii Equation

With the further research and development of the soliton phenomenon, the existence of solitary solutions has been proved in condensed matter physics. S-ince the successful realization of the experiment of Bose-Einstein condensation, many physicists and mathematicians are very interested in it. One oi the most popular mathematical model of depicting the Bose-Einstein condensation is the Gross-Pitaevskii equation(G-P equation). Inspired by the numerical sim-ulation of soliton initial state for soliton wave in laser transmission, which was studied by Chuanmiao Chen, et.al., this paper is mainly concerned with the soliton solutions of three kinds of G-P equation, whose potential is zero potential, optical lattice potential and harmonic potential respectively.Solitary wave is a special fluctuant form, how to imitate it numercially is very meaningful. In this paper, a Search-extension method(SEM) based on Chebyshev spectral collocation approach is adopted to calculate the soliton initial states of stationary Gross-Pitaevskii equation. Then, with these soliton initial states serving as the initial values, we shall discuss the initial-boundary problem of the original dynamical G-P equation and further investigate the basic properties of the solution. Finally, in order to study the time-dependent evolution of these soliton initial states, we propose a conservative discrete scheme, which combines the Chebyshev spectral collocation in space with re-laxation method in time. The numercial results demonstrate that these soliton initial states can produce solitary wave solutions.