Variational Approach to Study
PT-Symmetric Solitons in a Bose–Einstein Condensate with Non-locality of Interactions
Considering the non-locality of interactions in a Bose–Einstein condensate, the existence and stability of solitons subject to a - symmetric potential are discussed. In the framework of the variational approach, we investigate how the non-locality of interactions affects the self-localization and stability of a condensate with attractive two-body interactions. The results reveal that the non-locality of interactions dramatically influences the shape, width, and chemical potential of the condensate. Analytically variational computation also predicts that there exists a critical negative non-local interaction strength ( p c < 0) with each fixed two-body interaction ( g 0 < 0), and there exists no bright soliton solution for p 0 < p c . Furthermore, we study the effect of the non-locality interactions on the stability of the solitons using the Vakhitov–Kolokolov stability criterion. It is shown that for a positive non-local interaction ( p 0 > 0), there always exist stable bright solitons in some appropriate parameter regimes .
The dynamics of non-autonomous bright matter-wave solitons in Rabi coupled binary Bose–Einstein condensates is explored. By performing a unitary and similarity/lens-type transformation, we reduce the non-autonomous Gross–Pitaevskii (GP) equation into the celebrated Manakov model. Then, we construct the exact bright solitons of the non-autonomous GP system in the presence of time dependent nonlinearities for two specific forms, namely hyperbolic nonlinearities, which are of physical interest. The experimental possibilities of realizing the forms of temporally modulated potentials corresponding to these time dependent nonlinearities and supporting such localized structures are explored. Our study on the propagation of one soliton shows that the amplitude, velocity and shape of the bright soliton are altered by the time-dependent scattering length. We also analyze the non-trivial energy sharing collision of Manakov solitons in the presence of Rabi coupling and aforementioned nonlinearities. We find that breathers arise in two-soliton collisions and the nature of energy sharing collisions is altered from that of Manakov system due to Rabi coupling only. Further in the presence of time-dependent nonlinearities the collision scenario is again altered significantly. Finally, the stability of these localized structures is examined using a recently developed powerful analytic method by Quintero et al (2015 Phys. Rev. E 91 , 012905) and it is shown that the non-autonomous bright solitons are indeed stable. The evolution of position and velocity is also studied.
We present an analytical study on the dynamics of bright and dark solitons in Bose–Einstein condensates with time-varying atomic scattering length in a time-varying external parabolic potential. A set of exact soliton solutions of the one-dimensional Gross–Pitaevskii equation are obtained, including fundamental bright solitons, higher-order bright solitons, and dark solitons. The results show that the soliton's parameters (amplitude, width, and period) can be changed in a controllable manner by changing the scattering length and external potential. This may be helpful to design experiments.
Recent research works on ultra cold quantum gases demonstrated that dipolar Bose–Einstein condensates (BECs) exhibit rich spatiotemporal dynamic where both local and nonlocal interactions are considered. We explore theoretically the possibility of controlling the formation and dynamics of soliton molecules in binary dipolar condensates with spin-orbit coupling (SOC). We exploit the variational technique to derive the new equations of motion for the widths and amplitudes, the effective potential and the oscillation frequency of the molecules. Our study confirms the existence of stable localized bound states in an optical potential. We find that the integrity of the molecules is influenced by the physical parameters, notably the local and nonlocal interactions with the SOC. These parameters are carefully chosen by the Vakhitov–Kolokolov (VK) criterion to ensure the stability of the molecules. We present the results of numerical experiments and confirm the analytical predictions. Moreover, we show the soliton–soliton interaction in each molecule when the local interactions are strong.
We study the matter-wave solitons in Bose-Einstein condensate (BEC) trapped on a M\"obius st rip (MS), based on the respective Gross-Pitaevskii (GP) equation with the mean-field theory. In the linear regime, vortex states are characterized by quantum numbers,
n and
m, corresponding to the transverse and circumferential directions, with the phase structure determined by the winding number (WN)
m. Odd and even values of
n must associate, respectively, with integer and half-integer values of
m, the latter ones requiring two cycles of motion around MS for returning to the initial phase. Using variational and numerical methods, we solve the GP equation with the attractive nonlinearity, producing a family of ground-state (GS) solitons for values of the norm below the critical one, above which the collapse sets in. Vortex solitons with
n=1,m=1 and
n=2,m=1/2 are obtained in a numerical form. The vortex solitons with
n=1,m=1 are almost uniformly distributed in the azimuthal direction, while ones with
n=2,m=1/2 form localized states. The Vakhitov-Kolokolov criterion and linear-stability analysis for the GS soliton solutions and vortices with
n=1,m=1 demonstrates that they are completely stable, while the localized states with
n=2,m=1/2 are completely unstable. Finally, the motion of solitons on the MS and the collision of two solitons are discussed.
By means of variational solution and direct numerical simulation of the Gross-Pitaevskii equation (GPE), we have studied the stability of matter-wave solitons in two-dimensional (2D) Bose-Einstein condensations (BECs), with 2D linear and nonlinear optical lattices (OLs). Using the static variational approach and Vakhitov-Kolokolov criterion necessary for stability, we obtain the stability condition for solitons in different combinations of OL's parameters. We show that the 2D linear and nonlinear optical lattices allow us to stabilize 2D solitons for both attractive and repulsive interactions. We also study the time-evolution problems of 2D BECs, using the time-dependent variational approach and numerical solution of GPE for 2D linear and nonlinear OLs. Very good agreement between the results corresponding to both treatments is observed.