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.

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

2011 ◽  
Vol 110-116 ◽  
pp. 4492-4497
Author(s):  
Jun Lu
Keyword(s):  
Phase Space ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Einstein Condensate ◽  
Nonlinear Schrodinger Equation ◽  
Space Representation ◽  
Nonlinear Schrödinger ◽  
Bose Einstein Condensate ◽  
Bose Einstein

In this paper, we solve the rigorous solutions of attractive nonlinear Schrödinger equation which models the Bose-Einstein condensate, within the framework of the quantum phase space representation established by Torres-Vega and Frederick. By means of the “Fourier-like” projection transformation, we obtain the eigenfunctions in position and momentum spaces from the phase space eigenfunctions. As an example, we discuss the eigenfunction with a hypersecant part.

EXPECTED AND UNEXPECTED SOLUTIONS TO THE STATIONARY ONE-DIMENSIONAL NONLINEAR SCHRÖDINGER EQUATION

2001 ◽  
Vol 15 (10n11) ◽  
pp. 1663-1667
Author(s):  
LINCOLN D. CARR ◽  
CHARLES W. CLARK ◽  
WILLIAM P. REINHARDT
Keyword(s):  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Stationary Solutions ◽  
Einstein Condensate ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Bose Einstein Condensate ◽  
Dilute Gas

We present all stationary solutions to the nonlinear Schrödinger equation in one dimension for box and periodic boundary conditions. For both repulsive and attractive nonlinearity we find expected and unexpected solutions. Expected solutions are those that are in direct analogy with those of the linear Schödinger equation under the same boundary conditions. Unexpected solutions are those that have no such analogy. We give a physical interpretation for the unexpected solutions. We discuss the properties of all solution types and briefly relate them to experiments on the dilute-gas Bose-Einstein condensate.

scholarly journals CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS WITH AND WITHOUT HARMONIC POTENTIAL

2002 ◽  
Vol 12 (10) ◽  
pp. 1513-1523 ◽  
Author(s):  
RÉMI CARLES
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Space Dimension ◽  
Einstein Condensate ◽  
Nonlinear Schrodinger Equation ◽  
Harmonic Potential ◽  
Nonlinear Schrödinger ◽  
Change Of Variables ◽  
Bose Einstein Condensate

We use a change of variables that turns the critical nonlinear Schrödinger equation into the critical nonlinear Schrödinger equation with isotropic harmonic potential, in any space dimension. This change of variables is isometric on L2, and bijective on some time intervals. Using the known results for the critical nonlinear Schrödinger equation, this provides information for the properties of Bose–Einstein condensate in space dimension one and two. We discuss in particular the wave collapse phenomenon.

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.

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