Dynamics of Matter-Wave Solitons for Three-Dimensional Bose—Einstein Condensates with Time-Space Modulation

Abstract

By two direct assumption methods and symbolic computation, we present two families of one-soliton solutions and a family of two-soliton solutions with some arbitrary functions for the three-dimensional Gross—Pitaevskii equation with time-space modulation. Then we investigate the dynamics of these matter-wave solitons in three-dimensional Bose—Einstein condensates. We can see that the intensities of both one-solitons and two-solitons first increase rapidly to the condensation peak value, then decay very slowly to the background value. Thus these matter-wave solitons in three-dimensional Bose—Einstein condensates can remain for a sufficiently long time to be fully observed and modulated for real applications in today's experiments.

Novel exact solutions of coupled nonlinear Schrödinger equations with time—space modulation

“…( 34). It is necessary to point out such a fact that the number of the localized wave of ψ 2 is always twice that of ψ 1 (or exchange of both) in the expression of solutions (34). This differs from the situation in Refs.…”

“…( 32) belong to the stability region. In this way, γ(t) is a quasiperiodic solution and the solutions (34) show a quasiperiodic behavior. An example of this behavior is displayed in Fig.…”

“…with For the first example, we consider the periodic behavior of solutions (34), for which γ(t) occurs when ε = 0 or in the frontiers between the stability and instability regions of Eq. (32).…”

“…[19] However, the governing equations of fundamental theoretical interest are usually not completely integrable, as the varying coefficients do not satisfy the corresponding integrability conditions. These peculiar cases stimulate the researchers to find other powerful techniques that can be applied to deal with both the integrable and nonintegrable nonlinear wave equations, such as the similarity transformation method, [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] the Lie group symmetry method, [35][36][37][38] the Hirota bilinear method, [39,40] the subequation expansion method, [41][42][43][44] the F-expansion technique, [45,46] and the direct method. [47][48][49][50] In particular, the similarity transformation allows us to find self-similar solutions of equations by connecting the given variable-coefficient equation to the corresponding equation with constant coefficients.…”