Gross pitaevskii equation

Suppose that the initial data . In this chapter, we introduce the Gross-Pitaevskii equa- tion, and discuss its solutions, including vortices and solitons. In this Appendix we show in detail the derivation of the time-dependent Gross. Hamiltonian in terms of the Bose field . Collapse for attractive interaction.

Collective excitations of the condensate.

Thomas-Fermi approximation for repulsive interaction.

Small amplitude oscillations, Bogoliubov dispersion relation. Hydrodynamic equations, equation of continuity, Euler equation. Large amplitude oscillations, scaling . Institut für Theoretische Physik. Seminar in theoretical physics: Non-linear and non-hermitian quantum mechanics.

Bose-Einstein Condensation: Basics,. We also assume that the k-particle density matrices of the initial state are asymptotically factorized as N ! Identifying Фextreme parameter regimesХ, . In this context we also shortly review several approaches which allow, in principle, for calculating excited state solutions. It turns out that without modifications these are not applicable to strongly nonlinear . This generalization is obtained by applying the time-dependent variational principle to the variational manifold of continuous matrix product states.

In particular, we consider algorithms involving real and imaginary-time . We elucidate the steps needed to adapt to the magnetic case the proof of the derivation of the Gross-Pitaevskii . Abstract: We derive the Gross - Pitaevskii equation in two-dimension from the first principles of two-dimensional scattering theory. Moreover, the ground state energy and the . Provided the most recent approximation for the wave function is always used in the nonlinear atom-atom interaction potential energy, . We report solutions for different range of values for the repulsive and the . In this paper more general scalings shall be considered assuming . The SPGPE contains the challenge of both accurately evolving all modes in the low energy classical region of the system, and . Abstract: We prove that the Gross - Pitaevskii equation correctly describes the ground state energy and corresponding one-particle density matrix of rotating, dilute, trapped Bose gases with repulsive two-body interactions. Depending on the choice of the allowed free functions, the solutions can take the form of stationary dark or bright rings .