Manipulating localized matter waves in multicomponent Bose-Einstein condensates

Singularity in the matrix of the coupled Gross-Pitaevskii equations and the related state-transitions in three-species condensates

An approach is proposed to solve the coupled Gross-Pitaevskii equations (CGP) of the 3-species BEC in an analytical way under the Thomas-Fermi approximation (TFA). It was found that, when the strength of a kind of interaction increases and crosses over a critical value, a specific type of state-transition will occur and will cause a jump in the total energy. Due to the jump, the energy of the lowest symmetric state becomes considerably higher. This leaves a particular opportunity for the lowest asymmetric state to replace the symmetric states as the ground state. It was further found that the critical values are related to the singularity of either the matrix or a sub-matrix of the CGP. These critical values are not arising from the TFA but inherent in the CGP, and they can be analytically expressed. Furthermore, a model (in which two kinds of atoms separated from each other asymmetrically) has been proposed for the evaluation of the energy of the lowest asymmetric state. With this model the emergence of the asymmetric ground state is numerically confirmed under the TFA. The theoretical formalism of this paper is quite general and can be generalized for BEC with more than three species.

An analysis of spatiotemporal localized solutions in the variable coefficients (3 + 1)-dimensional nonlinear Schrödinger equation with six different forms of dispersion parameters

We construct spatiotemporal localized envelope solutions of a (3 + 1)-dimensional nonlinear Schrödinger equation with varying coefficients such as dispersion, nonlinearity and gain parameters through similarity transformation technique. The obtained localized rational solutions can serve as prototypes of rogue waves in different branches of science. We investigate the characteristics of constructed localized solutions in detail when it propagates through six different dispersion profiles, namely, constant, linear, Gaussian, hyperbolic, logarithm, and exponential. We also obtain expressions for the hump and valleys of rogue wave intensity profiles for these six dispersion profiles and study the trajectory of it in each case. Further, we analyze how the intensity of another localized solution, namely, breather, changes when it propagates through the aforementioned six dispersion profiles. Our studies reveal that these localized solutions co-exist with the collapsing solutions which are already found in the (3 + 1)-dimensional nonlinear Schrödinger equation. The obtained results will help to understand the corresponding localized wave phenomena in related fields.