460 research outputs found
Optimized explicit Runge-Kutta schemes for the spectral difference method applied to wave propagation problems
Explicit Runge-Kutta schemes with large stable step sizes are developed for
integration of high order spectral difference spatial discretization on
quadrilateral grids. The new schemes permit an effective time step that is
substantially larger than the maximum admissible time step of standard explicit
Runge-Kutta schemes available in literature. Furthermore, they have a small
principal error norm and admit a low-storage implementation. The advantages of
the new schemes are demonstrated through application to the Euler equations and
the linearized Euler equations.Comment: 37 pages, 3 pages of appendi
Pole dynamics for the Flierl-Petviashvili equation and zonal flow
We use a systematic method which allows us to identify a class of exact
solutions of the Flierl-Petvishvili equation. The solutions are periodic and
have one dimensional geometry. We examine the physical properties and find that
these structures can have a significant effect on the zonal flow generation.Comment: Latex 40 pages, seven figures eps included. Effect of variation of
g_3 is studied. New references adde
On the numerical evaluation of algebro-geometric solutions to integrable equations
Physically meaningful periodic solutions to certain integrable partial
differential equations are given in terms of multi-dimensional theta functions
associated to real Riemann surfaces. Typical analytical problems in the
numerical evaluation of these solutions are studied. In the case of
hyperelliptic surfaces efficient algorithms exist even for almost degenerate
surfaces. This allows the numerical study of solitonic limits. For general real
Riemann surfaces, the choice of a homology basis adapted to the
anti-holomorphic involution is important for a convenient formulation of the
solutions and smoothness conditions. Since existing algorithms for algebraic
curves produce a homology basis not related to automorphisms of the curve, we
study symplectic transformations to an adapted basis and give explicit formulae
for M-curves. As examples we discuss solutions of the Davey-Stewartson and the
multi-component nonlinear Schr\"odinger equations.Comment: 29 pages, 20 figure
Vortices in Bose-Einstein Condensates: Some Recent Developments
In this brief review we summarize a number of recent developments in the
study of vortices in Bose-Einstein condensates, a topic of considerable
theoretical and experimental interest in the past few years. We examine the
generation of vortices by means of phase imprinting, as well as via dynamical
instabilities. Their stability is subsequently examined in the presence of
purely magnetic trapping, and in the combined presence of magnetic and optical
trapping. We then study pairs of vortices and their interactions, illustrating
a reduced description in terms of ordinary differential equations for the
vortex centers. In the realm of two vortices we also consider the existence of
stable dipole clusters for two-component condensates. Last but not least, we
discuss mesoscopic patterns formed by vortices, the so-called vortex lattices
and analyze some of their intriguing dynamical features. A number of
interesting future directions are highlighted.Comment: 24 pages, 8 figs, ws-mplb.cls, to appear in Modern Physics Letters B
(2005
Bose-Einstein condensates in standing waves: The cubic nonlinear Schroedinger equation with a periodic potential
We present a new family of stationary solutions to the cubic nonlinear
Schroedinger equation with a Jacobian elliptic function potential. In the limit
of a sinusoidal potential our solutions model a dilute gas Bose-Einstein
condensate trapped in a standing light wave. Provided the ratio of the height
of the variations of the condensate to its DC offset is small enough, both
trivial phase and nontrivial phase solutions are shown to be stable. Numerical
simulations suggest such stationary states are experimentally observable.Comment: 4 pages, 4 figure
Modulated Amplitude Waves in Bose-Einstein Condensates
We analyze spatio-temporal structures in the Gross-Pitaevskii equation to
study the dynamics of quasi-one-dimensional Bose-Einstein condensates (BECs)
with mean-field interactions. A coherent structure ansatz yields a
parametrically forced nonlinear oscillator, to which we apply Lindstedt's
method and multiple-scale perturbation theory to determine the dependence of
the intensity of periodic orbits (``modulated amplitude waves'') on their wave
number. We explore BEC band structure in detail using Hamiltonian perturbation
theory and supporting numerical simulations.Comment: 5 pages, 4 figs, revtex, final form of paper, to appear in PRE
(forgot to include \bibliography command in last update, so this is a
correction of that; the bibliography is hence present again
Stability of Repulsive Bose-Einstein Condensates in a Periodic Potential
The cubic nonlinear Schr\"odinger equation with repulsive nonlinearity and an
elliptic function potential models a quasi-one-dimensional repulsive dilute gas
Bose-Einstein condensate trapped in a standing light wave. New families of
stationary solutions are presented. Some of these solutions have neither an
analog in the linear Schr\"odinger equation nor in the integrable nonlinear
Schr\"odinger equation. Their stability is examined using analytic and
numerical methods. All trivial-phase stable solutions are deformations of the
ground state of the linear Schr\"odinger equation. Our results show that a
large number of condensed atoms is sufficient to form a stable, periodic
condensate. Physically, this implies stability of states near the Thomas-Fermi
limit.Comment: 12 pages, 17 figure
The Unified Method: I Non-Linearizable Problems on the Half-Line
Boundary value problems for integrable nonlinear evolution PDEs formulated on
the half-line can be analyzed by the unified method introduced by one of the
authors and used extensively in the literature. The implementation of this
general method to this particular class of problems yields the solution in
terms of the unique solution of a matrix Riemann-Hilbert problem formulated in
the complex -plane (the Fourier plane), which has a jump matrix with
explicit -dependence involving four scalar functions of , called
spectral functions. Two of these functions depend on the initial data, whereas
the other two depend on all boundary values. The most difficult step of the new
method is the characterization of the latter two spectral functions in terms of
the given initial and boundary data, i.e. the elimination of the unknown
boundary values. For certain boundary conditions, called linearizable, this can
be achieved simply using algebraic manipulations. Here, we present an effective
characterization of the spectral functions in terms of the given initial and
boundary data for the general case of non-linearizable boundary conditions.
This characterization is based on the analysis of the so-called global
relation, on the analysis of the equations obtained from the global relation
via certain transformations leaving the dispersion relation of the associated
linearized PDE invariant, and on the computation of the large asymptotics
of the eigenfunctions defining the relevant spectral functions.Comment: 39 page
- …