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Physics Letters A

Exact soliton solutions of Gross Pitaevskii equation with a variable shape optical lattice potential

Z. Oztas, E. Kaplan

Published Sep 6, 2024• 7 citations
Publisher
DOI
10.1016/j.physleta.2024.129853

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Abstract We propose a theoretical framework, based on the two-component Gross–Pitaevskii equation (GPE), for the investigation of vortex solitons (VSs) in hybrid atomic–molecular Bose–Einstein condensates under the action of the stimulated Raman-induced photoassociation and square-optical-lattice potential. Stationary solutions of the coupled GPE system are obtained by means of the imaginary-time integration, while the temporal dynamics are simulated using the fourth-order Runge–Kutta algorithm. The analysis reveals stable rhombus-shaped VS shapes with topological charges m = 1 and 2 of the atomic component. The stability domains and spatial structure of these VSs are governed by three key parameters: the parametric-coupling strength ( χ ), atomic-molecular interaction strength ( g 12 ), and the optical-lattice potential depth ( V 0 ). By varying χ and g 12 , we demonstrate a structural transition where four-core rhombus-shaped VSs evolve into eight-core square-shaped modes, highlighting the nontrivial nonlinear dynamics of the system. This work establishes a connection between interactions of cold atoms and topologically structured matter waves in hybrid quantum systems.
Solitons in one-dimensional (1D) Bose-Einstein condensates with attractive interaction in variable anharmonic periodic potentials are presented. These variable periodic potentials allow us to consider a variety of potential shapes between the two special limits of the Kronig-Penney model and the inverse Kronig-Penney model, with the sinusoidal optical lattice as an intermediate case. The Gross-Pitaevskii equation in (1D) is taken as a model [53]. We have derived the stability conditions for BECs in variable anharmonic periodic potentials using the Vakhitov-Kolokolov criterion. We have found that the stability condition for solitons depends significantly on the shape parameter. In particular, depending on the values of the shape parameter, one and three localized states for the 1D case, respectively, have been found. We have shown that some of these localized states are stable or unstable. The stability conditions obtained are an extension of the existing results to: (1) the more general class of external potential, and (2) the Vakhitov-Kolokolov criterion has been generalized in the case of an external potential of arbitrary shape. A fully numerical simulation of the 1D Gross-Pitaevskii equation finally tests the results of the variational approximation. A good agreement between both the methods is observed.
We derive classes of exact solitonic solutions of the time-dependent Gross-Pitaevskii equation with repulsive and attractive interatomic interactions. The solutions correspond to a string of bright solitons with phase difference between adjacent solitons equal to πππ. While the relative phase, width, and distance between adjacent solitons turn out to be a constant of the motion, the center of mass of the string moves with a constant acceleration arising from the inhomogeneity of the background.