17 research outputs found
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
Fast soliton scattering by delta impurities
We study the Gross-Pitaevskii equation (nonlinear Schroedinger equation) with
a repulsive delta function potential. We show that a high velocity incoming
soliton is split into a transmitted component and a reflected component. The
transmitted mass (L^2 norm squared) is shown to be in good agreement with the
quantum transmission rate of the delta function potential. We further show that
the transmitted and reflected components resolve into solitons plus dispersive
radiation, and quantify the mass and phase of these solitons.Comment: 32 pages, 3 figure
Modulational Instability in Equations of KdV Type
It is a matter of experience that nonlinear waves in dispersive media,
propagating primarily in one direction, may appear periodic in small space and
time scales, but their characteristics --- amplitude, phase, wave number, etc.
--- slowly vary in large space and time scales. In the 1970's, Whitham
developed an asymptotic (WKB) method to study the effects of small
"modulations" on nonlinear periodic wave trains. Since then, there has been a
great deal of work aiming at rigorously justifying the predictions from
Whitham's formal theory. We discuss recent advances in the mathematical
understanding of the dynamics, in particular, the instability of slowly
modulated wave trains for nonlinear dispersive equations of KdV type.Comment: 40 pages. To appear in upcoming title in Lecture Notes in Physic
Bogoliubov sound speed in periodically modulated Bose-Einstein condensates
We study the Bogoliubov excitations of a Bose-condensed gas in an optical
lattice. Of primary interest is the long wavelength phonon dispersion for both
current-free and current-carrying condensates. We obtain the dispersion
relation by carrying out a systematic expansion of the Bogoliubov equations in
powers of the phonon wave vector. Our result for the current-carrying case
agrees with the one recently obtained by means of a hydrodynamic theory.Comment: 16 pages, no figure
Relativistic point dynamics and Einstein formula as a property of localized solutions of a nonlinear Klein-Gordon equation
Einstein's relation E=Mc^2 between the energy E and the mass M is the
cornerstone of the relativity theory. This relation is often derived in a
context of the relativistic theory for closed systems which do not accelerate.
By contrast, Newtonian approach to the mass is based on an accelerated motion.
We study here a particular neoclassical field model of a particle governed by a
nonlinear Klein-Gordon (KG) field equation. We prove that if a solution to the
nonlinear KG equation and its energy density concentrate at a trajectory, then
this trajectory and the energy must satisfy the relativistic version of
Newton's law with the mass satisfying Einstein's relation. Therefore the
internal energy of a localized wave affects its acceleration in an external
field as the inertial mass does in Newtonian mechanics. We demonstrate that the
"concentration" assumptions hold for a wide class of rectilinear accelerating
motions
On the Dynamics of solitons in the nonlinear Schroedinger equation
We study the behavior of the soliton solutions of the equation
i((\partial{\psi})/(\partialt))=-(1/(2m)){\Delta}{\psi}+(1/2)W_{{\epsilon}}'({\psi})+V(x){\psi}
where W_{{\epsilon}}' is a suitable nonlinear term which is singular for
{\epsilon}=0. We use the "strong" nonlinearity to obtain results on existence,
shape, stability and dynamics of the soliton. The main result of this paper
(Theorem 1) shows that for {\epsilon}\to0 the orbit of our soliton approaches
the orbit of a classical particle in a potential V(x).Comment: 29 page
Nonlinear coherent states and Ehrenfest time for Schrodinger equation
We consider the propagation of wave packets for the nonlinear Schrodinger
equation, in the semi-classical limit. We establish the existence of a critical
size for the initial data, in terms of the Planck constant: if the initial data
are too small, the nonlinearity is negligible up to the Ehrenfest time. If the
initial data have the critical size, then at leading order the wave function
propagates like a coherent state whose envelope is given by a nonlinear
equation, up to a time of the same order as the Ehrenfest time. We also prove a
nonlinear superposition principle for these nonlinear wave packets.Comment: 27 page
Bose-Einstein Condensates in Optical Lattices: Band-Gap Structure and Solitons
We analyze the existence and stability of spatially extended (Bloch-type) and
localized states of a Bose-Einstein condensate loaded into an optical lattice.
In the framework of the Gross-Pitaevskii equation with a periodic potential, we
study the band-gap structure of the matter-wave spectrum in both the linear and
nonlinear regimes. We demonstrate the existence of families of spatially
localized matter-wave gap solitons, and analyze their stability in different
band gaps, for both repulsive and attractive atomic interactions
On universality of critical behavior in the focusing nonlinear Schr\uf6dinger equation, elliptic umbilic catastrophe and the Tritronqu\ue9e solution to the Painlev\ue9-I equation
We argue that the critical behavior near the point of "gradient catastrophe" of the solution to the Cauchy problem for the focusing nonlinear Schrodinger equation i epsilon Psi(t) + epsilon(2)/2 Psi(xx) + vertical bar Psi vertical bar(2)Psi = 0, epsilon << 1, with analytic initial data of the form Psi( x, 0; epsilon) = A(x)e(i/epsilon) (S(x)) is approximately described by a particular solution to the Painleve-I equation
Stability of persistent currents in a Bose-Einstein condensate confined in a toroidal trap
Motivated by recent experiments in Bose-Einstein condensed atoms that have
been confined in toroidal traps, we examine the stability of persistent
currents in such systems. We investigate the extent that the stability of these
currents may be tunable, and the possible difficulties in their creation and
detection.Comment: 6 pages, 5 figure