4,021 research outputs found
A time-splitting pseudospectral method for the solution of the Gross-Pitaevskii equations using spherical harmonics with generalised-Laguerre basis functions
We present a method for numerically solving a Gross-Pitaevskii system of equations with a harmonic and a toroidal external potential that governs the dynamics of one- and two-component Bose-Einstein condensates. The method we develop maintains spectral accuracy by employing Fourier or spherical harmonics in the angular coordinates combined with generalised-Laguerre basis functions in the radial direction. Using an error analysis, we show that the method presented leads to more accurate results than one based on a sine transform in the radial direction when combined with a time-splitting method for integrating the equations forward in time. In contrast to a number of previous studies, no assumptions of radial or cylindrical symmetry is assumed allowing the method to be applied to 2D and 3D time-dependent simulations. This is accomplished by developing an efficient algorithm that accurately performs the generalised-Laguerre transforms of rotating Bose-Einstein condensates for different orders of the Laguerre polynomials. Using this spatial discretisation together with a second order Strang time-splitting method, we illustrate the scheme on a number of 2D and 3D computations of the ground state of a non-rotating and rotating condensate. Comparisons between previously derived theoretical results for these ground state solutions and our numerical computations show excellent agreement for these benchmark problems. The method is further applied to simulate a number of time-dependent problems including the Kelvin-Helmholtz instability in a two-component rotating condensate and the motion of quantised vortices in a 3D condensate
Fourth order real space solver for the time-dependent Schr\"odinger equation with singular Coulomb potential
We present a novel numerical method and algorithm for the solution of the 3D
axially symmetric time-dependent Schr\"odinger equation in cylindrical
coordinates, involving singular Coulomb potential terms besides a smooth
time-dependent potential. We use fourth order finite difference real space
discretization, with special formulae for the arising Neumann and Robin
boundary conditions along the symmetry axis. Our propagation algorithm is based
on merging the method of the split-operator approximation of the exponential
operator with the implicit equations of second order cylindrical 2D
Crank-Nicolson scheme. We call this method hybrid splitting scheme because it
inherits both the speed of the split step finite difference schemes and the
robustness of the full Crank-Nicolson scheme. Based on a thorough error
analysis, we verified both the fourth order accuracy of the spatial
discretization in the optimal spatial step size range, and the fourth order
scaling with the time step in the case of proper high order expressions of the
split-operator. We demonstrate the performance and high accuracy of our hybrid
splitting scheme by simulating optical tunneling from a hydrogen atom due to a
few-cycle laser pulse with linear polarization
Quadratic Mean Field Games
Mean field games were introduced independently by J-M. Lasry and P-L. Lions,
and by M. Huang, R.P. Malham\'e and P. E. Caines, in order to bring a new
approach to optimization problems with a large number of interacting agents.
The description of such models split in two parts, one describing the evolution
of the density of players in some parameter space, the other the value of a
cost functional each player tries to minimize for himself, anticipating on the
rational behavior of the others.
Quadratic Mean Field Games form a particular class among these systems, in
which the dynamics of each player is governed by a controlled Langevin equation
with an associated cost functional quadratic in the control parameter. In such
cases, there exists a deep relationship with the non-linear Schr\"odinger
equation in imaginary time, connexion which lead to effective approximation
schemes as well as a better understanding of the behavior of Mean Field Games.
The aim of this paper is to serve as an introduction to Quadratic Mean Field
Games and their connexion with the non-linear Schr\"odinger equation, providing
to physicists a good entry point into this new and exciting field.Comment: 62 pages, 4 figure
Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators
Dozens of exponential integration formulas have been proposed for the
high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries
and Ginzburg-Landau equations. We report the results of extensive comparisons
in MATLAB and Chebfun of such formulas in 1D, 2D and 3D, focusing on fourth and
higher order methods, and periodic semilinear stiff PDEs with constant
coefficients. Our conclusion is that it is hard to do much better than one of
the simplest of these formulas, the ETDRK4 scheme of Cox and Matthews
Y-System and Deformed Thermodynamic Bethe Ansatz
We introduce a new tool, the Deformed TBA (Deformed Thermodynamic Bethe
Ansatz), to analyze the monodromy problem of the cubic oscillator. The Deformed
TBA is a system of five coupled nonlinear integral equations, which in a
particular case reduces to the Zamolodchikov TBA equation for the 3-state Potts
model. Our method generalizes the Dorey-Tateo analysis of the (monomial) cubic
oscillator. We introduce a Y-system corresponding to the Deformed TBA and give
it an elegant geometric interpretation.Comment: 12 pages. Minor corrections in Section
Multi-Symplectic Simulation on Soliton-Collision for Nonlinear Perturbed Schrödinger Equation
Seeking solitary wave solutions and revealing their interactional characteristics for nonlinear evolution equations help us lot to comprehend the motion laws of the microparticles. As a local nonlinear dynamic behavior, the soliton-collision is difficult to be reproduced numerically. In this paper, the soliton-collision process in the nonlinear perturbed Schrödinger equation is simulated employing the multi-symplectic method. The multi-symplectic formulations are derived including the multi-symplectic form and three local conservation laws of the nonlinear perturbed Schrödinger equation. Employing the implicit midpoint rule, we construct a multi-symplectic scheme, which is equivalent to the Preissmann box scheme, for the nonlinear perturbed Schrödinger equation. The elegant structure-preserving properties of the multi-symplectic scheme are illustrated by the tiny maximum absolute residual of the discrete multi-symplectic structure at each time step in the numerical simulations. The effects of the perturbation strength on the soliton-collision in the nonlinear perturbed Schrödinger equation are reported in the numerical results in detail
Coupled transport in rotor models
Acknowledgement One of us (AP) wishes to acknowledge S. Flach for enlightening discussions about the relationship between the DNLS equation and the rotor model.Peer reviewedPublisher PD
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