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arXiv:cond-mat/9709268

A Unified Construction of Variational Methods for the Nonlinear Schroedinger Equation

Yeong E. Kim, Alexander L. Zubarev

Published Sep 24, 1997• 0 citations
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Abstract

Based on an approach introduced byGerjuoy, Rau, and Spruch, we constract variational principles in a systematic way for the nonlinear Schroedinger equation and obtain new variational principles for the case of Ginzburg-Pitaevskii-Gross equation (PGP) which is belived to describe accurately the Bose-Einstein condensation at zero temperature. As an application of these variational methods, a variational iteration method is proposed for calculating eigenvalue (chemical potential) and wave function for the GPG equation

DOI
10.48550/arxiv.cond-mat/9709268

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Based on an approach introduced byGerjuoy, Rau, and Spruch, we constract variational principles in a systematic way for the nonlinear Schroedinger equation and obtain new variational principles for the case of Ginzburg-Pitaevskii-Gross equation (PGP) which is belived to describe accurately the Bose-Einstein condensation at zero temperature. As an application of these variational methods, a variational iteration method is proposed for calculating eigenvalue (chemical potential) and wave function for the GPG equation
Based on an approach introduced by Gerjuoy, Rau, and Spruch, we construct variational principles in a systematic way for the nonlinear Schrödinger equation and obtain new variational principles for the case of Ginzburg-Pitaevskii-Gross equation (GPG) which is believed to describe accurately the Bose-Einstein condensation at zero temperature. As an application of these variational methods, a variational iteration method is proposed for calculating eigenvalue (chemical potential) and wave function for the GPG equation 1.
The variational method of coupled Gaussian functions is applied to Bose-Einstein condensates with long-range interactions. The time dependence of the condensate is described by dynamical equations for the variational parameters. We present the method and analytically derive the dynamical equations from the time-dependent Gross-Pitaevskii equation. The stability of the solutions is investigated using methods of nonlinear dynamics. The concept presented in this article will be applied to Bose-Einstein condensates with monopolar 1/r1/r1/r and dipolar 1/r31/{r}^{3}1/r3 interaction in the subsequent article [S. Rau et al., Phys. Rev. A 82, 023611 (2010)], where we will present a wealth of phenomena obtained using the ansatz with coupled Gaussian functions.
It is shown using the Gross–Pitaevskii equation that resonance states of Bose–Einstein condensates with attractive interactions can be stabilized into true bound states. A semiclassical variational approximation and an independent quantum variational numerical method are used to calculate the energies (chemical potentials) and linewidths of resonances of the time-independent Gross–Pitaevskii equation; both methods produce similar results. Borders between the regimes of resonances, bound states and, in two and three dimensions, collapse are identified.
Abstract. The motto of this paper is: Let's face Bose-Einstein condensation through nonUnear dynamics. We do this by choosing variational forms of the condensate wave functions (of given symmetry classes), which convert the Bose-Einstein condensates via the time-dependent Gross-Pitaevskii equation into Hamiltonian systems that can be studied using the methods of nonUnear dynamics. We consider in particular cold quantum gases where long-range interactions between the neutral atoms are present, in addition to the conventional short-range contact interaction, viz. gravity-Uke interactions, and dipole-dipole interactions. The residts obtained serve as a useful giude in the search for nonlinear dynamics effects in numerically exact quantum calcinations for Bose-Einstein condensates. A main residt is the prediction of the existence of stable islands as well as chaotic regions for excited states of dipolar condensates, which coidd be checked experimentally. Keywords: Bose-Einstein condensation, Gross-Pitaevskii equation, nonlinear dynamics effects PACS: 03.75.Kk, 34.20.Cf, 02.30.-f, 47.20.Ky 1.
We assume the macroscopic wave function of a Bose-Einstein condensate as a superposition of Gaussian wave packets, with time-dependent complex width parameters, insert it into the mean-field energy functional corresponding to the Gross-Pitaevskii equation (GPE) and apply the time-dependent variational principle. In this way the GPE is mapped onto a system of coupled equations of motion for the complex width parameters, which can be analyzed using the methods of nonlinear dynamics. We perform a stability analysis of the fixed points of the nonlinear system, and demonstrate that the eigenvalues of the Jacobian reproduce the low-lying quantum mechanical Bogoliubov excitation spectrum of a condensate in an axisymmetric trap.