237 research outputs found
Higher-order splitting algorithms for solving the nonlinear Schr\"odinger equation and their instabilities
Since the kinetic and the potential energy term of the real time nonlinear
Schr\"odinger equation can each be solved exactly, the entire equation can be
solved to any order via splitting algorithms. We verified the fourth-order
convergence of some well known algorithms by solving the Gross-Pitaevskii
equation numerically. All such splitting algorithms suffer from a latent
numerical instability even when the total energy is very well conserved. A
detail error analysis reveals that the noise, or elementary excitations of the
nonlinear Schr\"odinger, obeys the Bogoliubov spectrum and the instability is
due to the exponential growth of high wave number noises caused by the
splitting process. For a continuum wave function, this instability is
unavoidable no matter how small the time step. For a discrete wave function,
the instability can be avoided only for \dt k_{max}^2{<\atop\sim}2 \pi, where
.Comment: 10 pages, 8 figures, submitted to Phys. Rev.
Blow-up profile of rotating 2D focusing Bose gases
We consider the Gross-Pitaevskii equation describing an attractive Bose gas
trapped to a quasi 2D layer by means of a purely harmonic potential, and which
rotates at a fixed speed of rotation . First we study the behavior of
the ground state when the coupling constant approaches , the critical
strength of the cubic nonlinearity for the focusing nonlinear Schr{\"o}dinger
equation. We prove that blow-up always happens at the center of the trap, with
the blow-up profile given by the Gagliardo-Nirenberg solution. In particular,
the blow-up scenario is independent of , to leading order. This
generalizes results obtained by Guo and Seiringer (Lett. Math. Phys., 2014,
vol. 104, p. 141--156) in the non-rotating case. In a second part we consider
the many-particle Hamiltonian for bosons, interacting with a potential
rescaled in the mean-field manner w\int\_{\mathbb{R}^2} w(x) dx = 1\beta < 1/2a\_N \to a\_*N \to \infty$
Madelung, Gross-Pitaevskii and Korteweg
This paper surveys various aspects of the hydrodynamic formulation of the
nonlinear Schrodinger equation obtained via the Madelung transform in connexion
to models of quantum hydrodynamics and to compressible fluids of the Korteweg
type.Comment: 32 page
Recommended from our members
Anisotropic collapse in three-dimensional dipolar Bose-Einstein condensates
Spectral method for the time-dependent Gross-Pitaevskii equation with a harmonic trap
We study the numerical resolution of the time-dependent Gross-Pitaevskii
equation, a non-linear Schroedinger equation used to simulate the dynamics of
Bose-Einstein condensates. Considering condensates trapped in harmonic
potentials, we present an efficient algorithm by making use of a spectral
Galerkin method, using a basis set of harmonic oscillator functions, and the
Gauss-Hermite quadrature. We apply this algorithm to the simulation of
condensate breathing and scissors modes.Comment: 23 pages, 5 figure
Linear vs. nonlinear effects for nonlinear Schrodinger equations with potential
We review some recent results on nonlinear Schrodinger equations with
potential, with emphasis on the case where the potential is a second order
polynomial, for which the interaction between the linear dynamics caused by the
potential, and the nonlinear effects, can be described quite precisely. This
includes semi-classical regimes, as well as finite time blow-up and scattering
issues. We present the tools used for these problems, as well as their
limitations, and outline the arguments of the proofs.Comment: 20 pages; survey of previous result
Quantum Field Theoretical Description of Unstable Behavior of Trapped Bose-Einstein Condensates with Complex Eigenvalues of Bogoliubov-de Gennes Equations
The Bogoliubov-de Gennes equations are used for a number of theoretical works
on the trapped Bose-Einstein condensates. These equations are known to give the
energies of the quasi-particles when all the eigenvalues are real. We consider
the case in which these equations have complex eigenvalues. We give the
complete set including those modes whose eigenvalues are complex. The quantum
fields which represent neutral atoms are expanded in terms of the complete set.
It is shown that the state space is an indefinite metric one and that the free
Hamiltonian is not diagonalizable in the conventional bosonic representation.
We introduce a criterion to select quantum states describing the metastablity
of the condensate, called the physical state conditions. In order to study the
instability, we formulate the linear response of the density against the
time-dependent external perturbation within the regime of Kubo's linear
response theory. Some states, satisfying all the physical state conditions,
give the blow-up and damping behavior of the density distributions
corresponding to the complex eigenmodes. It is qualitatively consistent with
the result of the recent analyses using the time-dependent Gross-Pitaevskii
equation.Comment: 29 page
- …