Variational analysis of soliton scattering by external potentials

Umarov, Bakhram; Messikh, Azeddine; Regaa, Nassima; Baizakov, B. B.

doi:10.1088/1742-6596/435/1/012024

Cited by 13 publications

(16 citation statements)

References 18 publications

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“…which is written in the stationary frame of the soliton, with stationary coordinates (x, t). To accurately describe the perturbation dynamics, we choose a general variational ansatz of the form [71,83] ψ (x, t) ≡ a sech (…”

Section: B Variational Equationsmentioning

confidence: 99%

Stability of matter-wave solitons in a density-dependent gauge theory

Dingwall

Öhberg

2019

Phys. Rev. A

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We consider the linear stability of chiral matter-wave solitons described by a density-dependent gauge theory. By studying the associated Bogoliubov-de Gennes equations both numerically and analytically, we find that the stability problem effectively reduces to that of the standard Gross-Pitaevskii equation, proving that the solitons are stable to linear perturbations. In addition, we formulate the stability problem in the framework of the Vakhitov-Kolokolov criterion and provide supplementary numerical simulations which illustrate the absence of instabilities when the soliton is initially perturbed.

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Section: B Variational Equationsmentioning

confidence: 99%

Stability of matter-wave solitons in a density-dependent gauge theory

Dingwall

Öhberg

2019

Phys. Rev. A

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“…The Equation () can be derived from the following Lagrangian density:

scriptL = \frac{i}{2} ((), ψ {\dot{ψ}}^{*} - ψ^{*} \overset{ψ}{true}) + \frac{1}{2} {|ψ^{'}|}^{2} - V_{0} {sech}^{2} ((), x - italicut) {|ψ|}^{2} - \frac{g}{2} {|ψ|}^{4},

through the Euler–Lagrange equation:

\frac{\partial scriptL}{\partial ψ^{*}} - \frac{\partial}{\partial t} ((), \frac{\partial L}{\partial {\overset{ψ}{true}}^{*}}) - \frac{\partial}{\partial x} ((), \frac{\partial L}{\partial {ψ^{'}}^{*}}) = 0 .

The choice of the ansatz shape is very important [29]. In our case, a natural choice is the hyperbolic secant function [33] because at the linear limit of the 1D‐GPE it is possible to obtain the ground state solution of the one‐dimensional Schrödinger equation for the PT potential. Another plausible reason for choosing this ansatz is the fact that the hyperbolic secant function is an exact soliton solution (nonlinear solution) of the one‐dimensional nonlinear Schrödinger equation.…”

Section: Methodsmentioning

confidence: 99%

“…Now, in order to obtain the equations that govern the dynamic behavior of the variational parameters, we insert Equation () into Equation () and calculate the effective Lagrangian L by the average Lagrangian density:

L = (〈〉, L) = \int_{- \infty}^{+ \infty} scriptL italicdx .

Therefore, after integrating Lagrangian density over the spatial coordinate, we obtain the following effective Lagrangian:

L = N ([], \frac{π^{2}}{12} w^{2} \overset{c}{true} + \frac{π^{2}}{6} w^{2} c^{2} - \frac{{\overset{ξ}{true}}^{2}}{2} + \overset{δ}{true} + \frac{1}{6 w^{2}} - \frac{gN}{6 w} - \frac{V_{0}}{2 w} \int_{- \infty}^{+ \infty} {sech}^{2} ((), x - italicut) {sech}^{2} ((), \frac{x - ξ}{w}) italicdx) .

The evolution equations for variational parameters can be obtained from the Euler–Lagrange equation:

\frac{d}{dt} ((), \frac{\partial L}{\partial \dot{q}}) - \frac{\partial L}{\partial q} = 0,

where q are the generalized coordinates, q ≡ { δ , c , ξ , w }, related to variational parameters. The equation for the phase δ reduces to dN / dt = 0 and illustrates the conservation of the norm of the wave function or number of atoms in the BEC [33]. The other equations yield a set of coupled nonlinear integral equations:

\overset{w}{true} = 2 italiccw

\overset{ξ}{true} = \frac{V_{0}}{w^{2}} \int_{- \infty}^{+ \infty} {sech}^{2} ((), x - italicut) \tanh ((), \frac{x - ξ}{w})

…”

Section: Methodsmentioning

confidence: 99%

Dynamics and stability of matter‐wave solitons in cigar‐shaped Bose–Einstein condensates dragged by Pöschl–Teller potential

Pereira

Marangoni

Nascimento

2021

Int J of Quantum Chemistry

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In this paper, we theoretically investigate the dynamics and stability of matter‐wave solitons (MWS) in a cigar‐shaped Bose–Einstein condensate (BEC) consisting of atoms with attractive interatomic interactions trapped by Pöschl–Teller (PT) potential. In this scenario, we propose the idea of the BEC being dragged by PT potential at a constant speed. The BEC was modeled by the quasi‐one‐dimensional Gross–Piteaevskii Equation (1D‐GPE). In addition, an analytical (time‐dependent variational method) and numerical (split‐step Crank–Nicolson method) approach has been proposed to investigate the dynamic properties of solitons during the temporal evolution of this system. Both analytical and numerical results demonstrated that MWSs dragged by PT potential can be stable. From a general point of view, the results obtained in this paper can motivate both theoretical and experimental investigations on the dynamics and stability of solitons in dilute and ultracold unitary Bose gases trapped by PT potential, nonlinear optical properties in PT quantum wells, and propagation of solitons in nonlinear media.

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“…This method provides a set of ordinary differential equations describing the dynamics of system approximately. It can be used in BEC studies with several external potentials including harmonic potentials, optical lattice (OL) or time modulated potentials [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. It is well known that the collapse of condensate occurs when the atom number exists a critical value in the attractive nonlinearity regime.…”

Section: Introductionmentioning

confidence: 99%

“…Modulational instability analysis has been done for one-dimensional OL by taking higher order interactions into account [11]. Matter wave solutions like bright and dark solitons have been also investigated by the variational method [12][13][14][15]. Dynamics in disordered or random potentials is another topic of interest.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of Bose Einstein Condensate in Fourier Syntesized Optical Lattice Potential

Oztas

2019

Eskişehir Technical University Journal of Science and Technology a - Applied Sciences and Engineering

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In this study, we examine the dynamics of Bose Einstein condensate trapped by Fourier synthesized optical lattice potential. We use time-dependent variational method to find the ordinary differential equations of motion. We also solve Gross Pitaevskii equation numerically using split step Fourier method to verify our findings. Good agreement is achieved between the analytical and numerical results.

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Variational analysis of soliton scattering by external potentials

Cited by 13 publications

References 18 publications

Stability of matter-wave solitons in a density-dependent gauge theory

Stability of matter-wave solitons in a density-dependent gauge theory

Dynamics and stability of matter‐wave solitons in cigar‐shaped Bose–Einstein condensates dragged by Pöschl–Teller potential

Dynamics of Bose Einstein Condensate in Fourier Syntesized Optical Lattice Potential

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