Exact Chirped Soliton Solutions for the One-Dimensional Gross– Pitaevskii Equation with Time-Dependent Parameters

2012 ◽  
Vol 67 (3-4) ◽  
pp. 141-146 ◽  
Author(s):  
Zhenyun Qina ◽  
Gui Mu
Keyword(s):  
Soliton Solution ◽  
Soliton Solutions ◽  
Einstein Condensate ◽  
Time Dependent ◽  
Pitaevskii Equation ◽  
Zero Temperature ◽  
One Dimensional ◽  
Bose Einstein Condensate ◽  
The One ◽  
Bose Einstein

The Gross-Pitaevskii equation (GPE) describing the dynamics of a Bose-Einstein condensate at absolute zero temperature, is a generalized form of the nonlinear Schr¨odinger equation. In this work, the exact bright one-soliton solution of the one-dimensional GPE with time-dependent parameters is directly obtained by using the well-known Hirota method under the same conditions as in S. Rajendran et al., Physica D 239, 366 (2010). In addition, the two-soliton solution is also constructed effectively

scholarly journals Soliton Solutions and Collisions for the Multicomponent Gross–Pitaevskii Equation in Spinor Bose–Einstein Condensates

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Ming Wang ◽  
Guo-Liang He
Keyword(s):  
Soliton Solutions ◽  
Auxiliary Function ◽  
Einstein Condensate ◽  
Polarization Parameter ◽  
One Dimension ◽  
Pitaevskii Equation ◽  
Bose Einstein Condensate ◽  
The One ◽  
Bose Einstein ◽  
Two Peaks

In this paper, we investigate a five-component Gross–Pitaevskii equation, which is demonstrated to describe the dynamics of an F=2 spinor Bose–Einstein condensate in one dimension. By employing the Hirota method with an auxiliary function, we obtain the explicit bright one- and two-soliton solutions for the equation via symbolic computation. With the choice of polarization parameter and spin density, the one-soliton solutions are divided into four types: one-peak solitons in the ferromagnetic and cyclic states and one- and two-peak solitons in the polar states. For the former two, solitons share the similar shape of one peak in all components. Solitons in the polar states have the one- or two-peak profiles, and the separated distance between two peaks is inversely proportional to the value of polarization parameter. Based on the asymptotic analysis, we analyze the collisions between two solitons in the same and different states.

scholarly journals Stationary and Dynamical Solutions of the Gross-Pitaevskii Equation for a Bose-Einstein Condensate in a PT symmetric Double Well

10.14311/1797 ◽  
2013 ◽  
Vol 53 (3) ◽  
Author(s):  
Holger Cartarius ◽  
Dennis Dast ◽  
Daniel Haag ◽  
Günter Wunner ◽  
Rüdiger Eichler ◽  
...  
Keyword(s):  
Wave Function ◽  
Three Dimensional ◽  
Stationary Solutions ◽  
Einstein Condensate ◽  
Time Dependent ◽  
Pitaevskii Equation ◽  
One Dimensional ◽  
Bose Einstein Condensate ◽  
Bose Einstein ◽  
Double Well Potential

We investigate the Gross-Pitaevskii equation for a Bose-Einstein condensate in a PT symmetric double-well potential by means of the time-dependent variational principle and numerically exact solutions. A one-dimensional and a fully three-dimensional setup are used. Stationary states are determined and the propagation of wave function is investigated using the time-dependent Gross-Pitaevskii equation. Due to the nonlinearity of the Gross-Pitaevskii equation the potential dependson the wave function and its solutions decide whether or not the Hamiltonian itself is PT symmetric. Stationary solutions with real energy eigenvalues fulfilling exact PT symmetry are found as well as PT broken eigenstates with complex energies. The latter describe decaying or growing probability amplitudes and are not true stationary solutions of the time-dependent Gross-Pitaevskii equation. However, they still provide qualitative information about the time evolution of the wave functions.

Dark–dark solitons for the coupled spatially modulated Gross–Pitaevskii system in the Bose–Einstein condensation

2020 ◽  
Vol 34 (26) ◽  
pp. 2050282 ◽  
Author(s):  
Xin Zhao ◽  
Bo Tian ◽  
Qi-Xing Qu ◽  
Yu-Qiang Yuan ◽  
Xia-Xia Du ◽  
...  
Keyword(s):  
Soliton Solutions ◽  
Dark Soliton ◽  
Einstein Condensate ◽  
Einstein Condensation ◽  
Time Coordinate ◽  
Elastic Interactions ◽  
Bose Einstein Condensate ◽  
The One ◽  
Bose Einstein ◽  
Spatially Varying

Investigation in this paper is the spatially modulated two-component GP system with Rabi coupling in a Bose–Einstein condensate consisting of the two hyperfine states. Based on the Kadomtsev–Petviashvili hierarchy reduction, we derive the Gramian expression of the one- and two-dark–dark soliton solutions. The nonlinearity coefficients [Formula: see text] and the external spatially varying trapping potential [Formula: see text] can be constrained as the functions of [Formula: see text], where [Formula: see text] is the spatial coordinate, [Formula: see text] is the time coordinate, [Formula: see text] is the dispersion parameter. With the Rabi coupling coefficient [Formula: see text] increasing, period along [Formula: see text] decreases. When [Formula: see text] is a constant, soliton propagates stably with the amplitude and velocity unvarying; When [Formula: see text] is a function of [Formula: see text], background is periodic and velocity of the soliton varies with [Formula: see text] increasing. Head-on and overtaking elastic interactions between the two solitons are presented analytically and graphically.

Sonic horizon formation for coupled one-dimensional Bose–Einstein condensate in an elongated harmonic potential

2018 ◽  
Vol 32 (29) ◽  
pp. 1850352
Author(s):  
Ying Wang ◽  
Shuyu Zhou
Keyword(s):  
Expansion Method ◽  
Einstein Condensate ◽  
Wave Functions ◽  
Harmonic Potential ◽  
Pitaevskii Equation ◽  
Two Component System ◽  
One Dimensional ◽  
Bose Einstein Condensate ◽  
Two Component ◽  
Bose Einstein

We theoretically studied the sonic horizon formation problem for coupled one-dimensional Bose–Einstein condensate trapped in an external elongated harmonic potential. Based on the coupled (1[Formula: see text]+[Formula: see text]1)-dimensional Gross–Pitaevskii equation and F-expansion method under Thomas–Fermi formulation, we derived analytical wave functions of a two-component system, from which the sonic horizon’s occurrence criteria and location were derived and graphically demonstrated. The theoretically derived results of sonic horizon formation agree pretty well with that from the numerically calculated values.

scholarly journals A Convergent Three-Step Numerical Method to Solve a Double-Fractional Two-Component Bose–Einstein Condensate

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1412
Author(s):  
Adán J. Serna-Reyes ◽  
Jorge E. Macías-Díaz ◽  
Nuria Reguera
Keyword(s):  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Einstein Condensate ◽  
One Dimensional ◽  
Bose Einstein Condensate ◽  
Nonlinear Parabolic ◽  
Complex Functions ◽  
Two Component ◽  
The One ◽  
Bose Einstein

This manuscript introduces a discrete technique to estimate the solution of a double-fractional two-component Bose–Einstein condensate. The system consists of two coupled nonlinear parabolic partial differential equations whose solutions are two complex functions, and the spatial fractional derivatives are interpreted in the Riesz sense. Initial and homogeneous Dirichlet boundary data are imposed on a multidimensional spatial domain. To approximate the solutions, we employ a finite difference methodology. We rigorously establish the existence of numerical solutions along with the main numerical properties. Concretely, we show that the scheme is consistent in both space and time as well as stable and convergent. Numerical simulations in the one-dimensional scenario are presented in order to show the performance of the scheme. For the sake of convenience, A MATLAB code of the numerical model is provided in the appendix at the end of this work.

STABILITY OF BOSE–EINSTEIN CONDENSATES IN A LORENTZ VIOLATING SCENARIO

2010 ◽  
Vol 25 (06) ◽  
pp. 459-469 ◽  
Author(s):  
E. CASTELLANOS ◽  
A. CAMACHO
Keyword(s):  
Nonlinear Term ◽  
Scattering Length ◽  
Lorentz Violation ◽  
Einstein Condensate ◽  
Pitaevskii Equation ◽  
Quantum Structure ◽  
Bose Einstein Condensate ◽  
The Stability ◽  
The One ◽  
Bose Einstein

We analyze the stability of a Bose–Einstein condensate in a Lorentz violating scenario, which is characterized by a deformation in the dispersion relation. The incorporation of a Lorentz violation within the bosonic statistics has, as a consequence, the emergence of a pseudo-interaction, the one can be associated to a characteristic scattering length. In addition, we calculate the relevant parameters associated to the stability of such condensate incorporating this pseudo-interaction in the nonlinear term of the Gross–Pitaevskii equation. We show that these parameters must be corrected, as a consequence of the quantum structure of spacetime.

scholarly journals Parafermionic behavior of Bose-Einstein condensates

2019 ◽  
Vol 21 ◽  
pp. 71
Author(s):  
A. Martinou ◽  
D. Bonatsos
Keyword(s):  
Optical Potential ◽  
Optical Trap ◽  
Einstein Condensate ◽  
Pitaevskii Equation ◽  
One Dimensional ◽  
Bright Solitons ◽  
Bose Einstein Condensate ◽  
Repulsive Interactions ◽  
Bose Einstein

Bright solitons of 7Li atoms in a quasi one-dimensional optical trap, formed in a stable Bose–Einstein condensate in which the interactions have been magnetically tuned from repulsive to attractive, have been seen to exhibit repulsive interactions among them when set in motion by offsetting the optical potential. Solving first the Gross–Pitaevskii equation for the special conditions corresponding to the experiment, we show then that this system can be described in terms of generalized parafermionic oscillators, the order of the parafermions being related to the strength of the interaction among the atoms and being a measure of the bosonic behavior vs. the fermionic behavior of the system.

scholarly journals Emission of Solitons From an Obstacle Moving in the Bose-Einstein Condensate

2021 ◽  
Vol 9 ◽  
Author(s):  
Yu Song ◽  
Yu Mo ◽  
Shiping Feng ◽  
Shi-Jie Yang
Keyword(s):  
Numerical Simulations ◽  
Einstein Condensate ◽  
Pitaevskii Equation ◽  
Nonlinear Interactions ◽  
One Dimensional ◽  
System Parameters ◽  
Dark Solitons ◽  
Bose Einstein Condensate ◽  
Bose Einstein ◽  
Moving Obstacle

Dark solitons dynamically generated from a potential moving in a one-dimensional Bose-Einstein condensate are displayed. Based on numerical simulations of the Gross-Pitaevskii equation, we find that the moving obstacle successively emits a series of solitons which propagate at constant speeds. The dependence of soliton emission on the system parameters is examined. The formation mechanism of solitons is interpreted as interference between a diffusing wavepacket and the condensate background, enhanced by the nonlinear interactions.PACS numbers: 03.75.Mn, 03.75.Lm, 05.30.Jp

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