scholarly journals The Darboux Transformation and N-Soliton Solutions of Coupled Cubic-Quintic Nonlinear Schrödinger Equation on a Time-Space Scale

2021 ◽  
Vol 6 (1) ◽  
pp. 12
Author(s):  
Huanhe Dong ◽  
Chunming Wei ◽  
Yong Zhang ◽  
Mingshuo Liu ◽  
Yong Fang
Keyword(s):  
Dynamic System ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Einstein Condensate ◽  
Nonlinear Schrödinger ◽  
Bose Einstein Condensate ◽  
Space Scale ◽  
Time Space ◽  
Bose Einstein ◽  
Quintic Nonlinear Schrödinger Equation

The coupled cubic-quintic nonlinear Schrödinger (CQNLS) equation is a universal mathematical model describing many physical situations, such as nonlinear optics and Bose–Einstein condensate. In this paper, in order to simplify the process of similar analysis with different forms of the coupled CQNLS equation, this dynamic system is extended to a time-space scale based on the Lax pair and zero curvature equation. Furthermore, Darboux transformation of the coupled CQNLS dynamic system on a time-space scale is constructed, and the N-soliton solution is obtained. These results effectively combine the theory of differential equations with difference equations and become a bridge connecting continuous and discrete analysis.

scholarly journals On the Existence of Bright Solitons in Cubic-Quintic Nonlinear Schrödinger Equation with Inhomogeneous Nonlinearity

2008 ◽  
Vol 2008 ◽  
pp. 1-14 ◽  
Author(s):  
Juan Belmonte-Beitia
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Bifurcation Theory ◽  
Schrodinger Equation ◽  
Chemical Potential ◽  
Soliton Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Bose Einstein ◽  
Quintic Nonlinear Schrödinger Equation

We give a proof of the existence of stationary bright soliton solutions of the cubic-quintic nonlinear Schrödinger equation with inhomogeneous nonlinearity. By using bifurcation theory, we prove that the norm of the positive solution goes to zero as the parameterλ, called chemical potential in the Bose-Einstein condensates' literature, tends to zero. Moreover, we solve the time-dependent cubic-quintic nonlinear Schrödinger equation with inhomogeneous nonlinearities by using a numerical method.

Integrability and Multi-Soliton Solutions for a Variable-Coefficient Coupled Gross–Pitaevskii System for Atomic MatterWaves

2012 ◽  
Vol 67 (10-11) ◽  
pp. 525-533
Author(s):  
Zhi-Qiang Lin ◽  
Bo Tian ◽  
Ming Wang ◽  
Xing Lu
Keyword(s):  
Variable Coefficient ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Einstein Condensate ◽  
Matter Wave ◽  
Bilinear Method ◽  
Matter Waves ◽  
Bose Einstein Condensate ◽  
Bose Einstein ◽  
Hirota Bilinear

Under investigation in this paper is a variable-coefficient coupled Gross-Pitaevskii (GP) system, which is associated with the studies on atomic matter waves. Through the Painlev´e analysis, we obtain the constraint on the variable coefficients, under which the system is integrable. The bilinear form and multi-soliton solutions are derived with the Hirota bilinear method and symbolic computation. We found that: (i) in the elastic collisions, an external potential can change the propagation of the soliton, and thus the density of the matter wave in the two-species Bose-Einstein condensate (BEC); (ii) in the shape-changing collision, the solitons can exchange energy among different species, leading to the change of soliton amplitudes.We also present the collisions among three solitons of atomic matter waves.

On the soliton solutions of the spinor Bose-Einstein condensate

2007 ◽  
Author(s):  
N. A. Kostov ◽  
V. A. Atanasov ◽  
V. S. Gerdjikov ◽  
G. G. Grahovski
Keyword(s):  
Soliton Solutions ◽  
Einstein Condensate ◽  
Bose Einstein Condensate ◽  
Bose Einstein

scholarly journals Nonperiodic Oscillation of Bright Solitons in Condensates with a Periodically Oscillating Harmonic Potential 10.5560/ZNA.2012-0085

2012 ◽  
Vol 67 (12) ◽  
pp. 723-728
Author(s):  
Zhang-Ming He ◽  
Deng-Long Wang ◽  
Yan-Chao She ◽  
Jian-Wen Ding ◽  
Xiao-Hong Yan
Keyword(s):  
Darboux Transformation ◽  
Oscillation Frequency ◽  
Dynamic Properties ◽  
Einstein Condensate ◽  
Harmonic Potential ◽  
Bright Solitons ◽  
Bose Einstein Condensate ◽  
Oscillating Potential ◽  
Bose Einstein

Considering a periodically oscillating harmonic potential, we explored the dynamic properties of bright solitons in a Bose-Einstein condensate by using Darboux transformation. It is found that the soliton movement exhibits a nonperiodic oscillation under a slow oscillating potential, while it is hardly affected under a fast oscillating potential. Furthermore, the head-on and/or ‘chase’ collisions of two solitons have been obtained, which could be controlled by the oscillation frequency of the potential.

Repulsive Nonlinear Schrödinger Equation and Bose-Einstein Condensate in Phase Space

2011 ◽  
Vol 403-408 ◽  
pp. 132-137
Author(s):  
Jun Lu ◽  
Yun Zhi Wang ◽  
Xiao Yun Mu
Keyword(s):  
Phase Space ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Einstein Condensate ◽  
Nonlinear Schrodinger Equation ◽  
Space Representation ◽  
Nonlinear Schrödinger ◽  
Bose Einstein Condensate ◽  
Bose Einstein ◽  
Phase Space Representation

Within the framework of the quantum phase-space representation established by Torres-Vega and Frederick, the rigorous solutions of repulsive nonlinear Schrödinger equation are solved, which models the dilute-gas Bose-Einstein condensate. The eigenfunctions in position and momentum spaces can be obtained through the “Fourier-like” projection transformation from the phase-space eigenfunctions. It shows that the wave-mechanics method in the phase-space representation could be extended to the nonlinear Schrödinger equations. The research provides the foundation for the approximate calculation in future.

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.

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.

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