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Journal of Physics B Atomic Molecular and Optical Physics

Modulational instability and soliton control in a cubic–quintic dissipative Gross–Pitaevskii equation with distributed coefficients*

Emmanuel Kengne, Wu‐Ming Liu

Published Aug 28, 2020• 7 citations
Publisher

Abstract

Abstract In this work, we consider the generalized cubic–quintic dissipative Gross–Pitaevskii equation, which governs the dynamics of matter wave solitons in Bose–Einstein condensates with two- and three-body interatomic interactions in a spatiotemporal-dependent dissipative potential consisting of parabolic, linear, and complex terms. By using the ansatz method, the modulational instability and gray, kink, and bright soliton solutions are presented under certain parametric conditions. We found that the complex potential, related to the feeding or the loss of atoms by the condensates seriously modifies the instability and stability domain, while the linear potential has not effect on the stability of the system. With the use of exact analytical soliton-like solutions, we investigate analytically the solitons control system, and the results show that the soliton control system may relax the limitations to parametric conditions. We found that the motion of matter wave solitons in the systems can be manipulated by controlling both the external harmonic and linear trapping potentials. We have established that the amplitude of the matter wave solitons keep no change in propagating in the system though the total number of the condensate atoms decreases (increases) when the condensate losses (gains) atoms. We also showed that the three-body interatomic interactions is responsible of the soliton compression. Our results also revealed that the found exact soliton-like solutions can be used to describe the compression of matter wave solitons in BEC system with loss of atoms.

DOI
10.1088/1361-6455/abb3ae

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Abstract In this work, we consider the generalized cubic–quintic dissipative Gross–Pitaevskii equation, which governs the dynamics of matter wave solitons in Bose–Einstein condensates with two- and three-body interatomic interactions in a spatiotemporal-dependent dissipative potential consisting of parabolic, linear, and complex terms. By using the ansatz method, the modulational instability and gray, kink, and bright soliton solutions are presented under certain parametric conditions. We found that the complex potential, related to the feeding or the loss of atoms by the condensates seriously modifies the instability and stability domain, while the linear potential has not effect on the stability of the system. With the use of exact analytical soliton-like solutions, we investigate analytically the solitons control system, and the results show that the soliton control system may relax the limitations to parametric conditions. We found that the motion of matter wave solitons in the systems can be manipulated by controlling both the external harmonic and linear trapping potentials. We have established that the amplitude of the matter wave solitons keep no change in propagating in the system though the total number of the condensate atoms decreases (increases) when the condensate losses (gains) atoms. We also showed that the three-body interatomic interactions is responsible of the soliton compression. Our results also revealed that the found exact soliton-like solutions can be used to describe the compression of matter wave solitons in BEC system with loss of atoms.
To analytically investigate the matter-wave solitons of Bose–Einstein condensates (BECs) in time-dependent complex potential, we consider a cubic-quintic Gross–Pitaevskii (GP) equation with distributed coefficients and a dissipative term. By introducing a suitable ansatz, we establish the criterion of the modulational instability (MI) of the system and present an explicit expression for the growth rate of a purely growing MI. Effects of the parabolic background potential, as well as of the linear potential, the gain/loss parameter, and the two- and three-body interatomic interactions on the MI are investigated. We show how the feeding/loss parameter can be well used to control the instability of the system. The analytical resolution of the considered GP equation leads to exact bright, dark and kink solitary wave solutions which are used to investigate analytically the dynamics of matter-wave solitons in BECs under consideration. These analytical investigations show that the amplitude and the motion of bright, dark and kink solitary waves depend on the strengths of the two- and three-body interatomic interactions, as well as on the strengths of the external trapping potential and the parameter of the gain/loss of atoms in the condensate.
In this paper, we consider a Gross-Pitaevskii (GP) equation with a time-dependent nonlinearity and a spatiotemporal complex linear term which describes the dynamics of matter-wave solitons in Bose-Einstein condensates (BECs) with time-dependent interatomic interactions in a parabolic potential in the presence of feeding or loss of atoms. We establish the integrability conditions under which analytical solutions describing the modulational instability and the propagation of both bright and dark solitary waves on a continuous wave background are obtained. The obtained integrability conditions also appear as the conditions under which the solitary waves of the BECs can be managed by controlling the functional gain or loss parameter. For specific BECs, the dynamics of bright and dark solitons are investigated analytically through the found exact solutions of the GP equation. Our results show that under the integrability conditions, the gain or loss parameter of the GP equation can be used to manage the motion of both bright and dark solitons. We show that for BECs with loss (gain) of atoms, the bright and dark solitons during their propagation have a compression (broadening) in their width. Furthermore, under a safe range of parameters and under the integrability conditions, it is possible to squeeze a bright soliton of BECs with loss of atoms into the assumed peak matter density, which can provide an experimental tool for investigating the range of validity of the 1D GP equation. Our results also reveal that under the conditions of the solitary wave management, neither the injection or the ejection of atoms from the condensate affects the soliton peak during its propagation.
A one-dimensional (1D) Gross-Pitaevskii (GP) equation which governs the dynamics of Bose-Einstein condensates (BECs) with time-dependent scattering length in a harmonic potential is considered. Through a modified phase imprint transformation into the macroscopic order parameter, we suitably generate a derivative cubic-quintic Gross-Pitaevskii (DCQ-GP) equation that describes the dynamics of condensates in the harmonic potential for both twoand three-body interatomic interactions in the presence of nonlinear dispersion. Employing a modified lens-type transformation, we reduce the DCQ-GP equation into a derivative cubic-quintic Ginzburg-Landau (DCQ-GL) equation that allows us to investigate the modulational instability (MI) of the system and study, through the analytical exact solutions of the considered 1D GP equation, the effects of the three-body interactions and nonlinear dispersion on the dynamics of the BECs. We observe that for BECs with repulsive two-body interatomic interactions, the attractive three-body interactions and the nonlinear dispersion have opposite effects on the condensate: (i) The attractive three-body interactions enhance the instability, while the nonlinear dispersion softens the instability of the system; (ii) the attractive three-body interactions contributes to the ejections of atoms from the condensates while the nonlinear dispersion contributes to the accumulation of atoms in the condensates; (iii) the attractive three-body interactions contributes to the broadening of the matter wave solitons, while the nonlinear dispersion contributes to the compression of the matter wave solitons.
Beyond the mean-field theory, a new model of the Gross–Pitaevskii equation (GPE) that describes the dynamics of Bose–Einstein condensates (BECs) is derived using an appropriate phase-imprint on the old wavefunction. This modified version of the GPE in addition to the two-body interactions term, also takes into account effects of the three-body interactions. The three-body interactions consist of a quintic term and the delayed nonlinear response of the condensate system term. Then, the modulational instability (MI) of the new GPE confined in an attractive harmonic potential is investigated. The analytical study shows that the three-body interactions destabilize more the condensate system while the external potential alleviates the instability. Numerical results confirm the theoretical predictions. Further numerical investigations of the behavior of solitons reveal that the three-body interactions enhance the appearance of solitons, increase the number of solitons generated and deeply change the lifetime of solitons. Moreover, the external potential delays the appearance of solitons. Besides, a new initial condition is introduced which enables to increase the number of solitons created and deeply affects the trail of chains of solitons generated. Moreover, the MI of a condensate without the external potential, and in a repulsive potential is also investigated.
The matter-wave solutions of Bose-Einstein condensates with three-body interaction are examined through the one-dimensional Gross-Pitaevskii equation. By using a modified lens-type transformation and a further extension of the tanh-function method we obtain the exact analytical solutions which describe the propagation of kink-shaped solitons, anti-kink-shaped solitons, and other families of solitary waves. We realize that the shape of a kink solitary wave depends on both the scattering length and the parameter of atomic exchange with the substrate. The stability of the solitary waves is examined using analytical and numerical methods. Our results can also be applied to nonlinear optics in the presence of cubic-quintic media.
We consider the dynamics of dark matter wave solitons moving through non-uniform cigar-shaped Bose–Einstein condensates described by the mean field Gross–Pitaevskii equation with generalized nonlinearities, in the case when the condition for the modulation stability of the Bose–Einstein condensate is fulfilled. The analytical expression for the frequency of the oscillations of a deep dark soliton is derived for nonlinearities which are arbitrary functions of the density, while specific results are discussed for the physically relevant case of a cubic–quintic nonlinearity modelling two- and three-body interactions, respectively. Opposite to the usual (cubic) Gross–Pitaevskii equation for which the dark soliton effective mass is known to be constant (equal to 2), in the presence of a cubic–quintic nonlinearity we find that the effective mass depends on the product of the initial density background and the ratio between the coefficient of quintic and cubic nonlinearities, this leading to the interesting possibility of measuring three-body interactions directly from the dark soliton dynamics. A comparison between analytical results and direct numerical simulations of the cubic–quintic Gross–Pitaevskii equation shows good agreement between them which confirms the validity of our approach.