243 research outputs found
Bifurcation of Nonlinear Bloch Waves from the Spectrum in the Gross-Pitaevskii Equation
We rigorously analyze the bifurcation of stationary so called nonlinear Bloch
waves (NLBs) from the spectrum in the Gross-Pitaevskii (GP) equation with a
periodic potential, in arbitrary space dimensions. These are solutions which
can be expressed as finite sums of quasi-periodic functions, and which in a
formal asymptotic expansion are obtained from solutions of the so called
algebraic coupled mode equations. Here we justify this expansion by proving the
existence of NLBs and estimating the error of the formal asymptotics. The
analysis is illustrated by numerical bifurcation diagrams, mostly in 2D. In
addition, we illustrate some relations of NLBs to other classes of solutions of
the GP equation, in particular to so called out--of--gap solitons and truncated
NLBs, and present some numerical experiments concerning the stability of these
solutions.Comment: 32 pages, 12 figures, changes: discussion of assumptions reorganized,
a new section on stability of the studied solutions, 15 new references adde
Vortex families near a spectral edge in the Gross-Pitaevskii equation with a two-dimensional periodic potential
We examine numerically vortex families near band edges of the Bloch wave
spectrum in the Gross--Pitaevskii equation with a two-dimensional periodic
potential and in the discrete nonlinear Schroedinger equation. We show that
besides vortex families that terminate at a small distance from the band edges
via fold bifurcations there exist vortex families that are continued all way to
the band edges.Comment: 12 pages, 8 figure
Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential
We address a two-dimensional nonlinear elliptic problem with a
finite-amplitude periodic potential. For a class of separable symmetric
potentials, we study the bifurcation of the first band gap in the spectrum of
the linear Schr\"{o}dinger operator and the relevant coupled-mode equations to
describe this bifurcation. The coupled-mode equations are derived by the
rigorous analysis based on the Fourier--Bloch decomposition and the Implicit
Function Theorem in the space of bounded continuous functions vanishing at
infinity. Persistence of reversible localized solutions, called gap solitons,
beyond the coupled-mode equations is proved under a non-degeneracy assumption
on the kernel of the linearization operator. Various branches of reversible
localized solutions are classified numerically in the framework of the
coupled-mode equations and convergence of the approximation error is verified.
Error estimates on the time-dependent solutions of the Gross--Pitaevskii
equation and the coupled-mode equations are obtained for a finite-time
interval.Comment: 32 pages, 16 figure
Nonlinear Schr\"odinger equation for a PT symmetric delta-functions double well
The time-independent nonlinear Schr\"odinger equation is solved for two
attractive delta-function shaped potential wells where an imaginary loss term
is added in one well, and a gain term of the same size but with opposite sign
in the other. We show that for vanishing nonlinearity the model captures all
the features known from studies of PT symmetric optical wave guides, e.g., the
coalescence of modes in an exceptional point at a critical value of the
loss/gain parameter, and the breaking of PT symmetry beyond. With the
nonlinearity present, the equation is a model for a Bose-Einstein condensate
with loss and gain in a double well potential. We find that the nonlinear
Hamiltonian picks as stationary eigenstates exactly such solutions which render
the nonlinear Hamiltonian itself PT symmetric, but observe coalescence and
bifurcation scenarios different from those known from linear PT symmetric
Hamiltonians.Comment: 16 pages, 9 figures, to be published in Journal of Physics
Quasiperiodic Dynamics in Bose-Einstein Condensates in Periodic Lattices and Superlattices
We employ KAM theory to rigorously investigate quasiperiodic dynamics in
cigar-shaped Bose-Einstein condensates (BEC) in periodic lattices and
superlattices. Toward this end, we apply a coherent structure ansatz to the
Gross-Pitaevskii equation to obtain a parametrically forced Duffing equation
describing the spatial dynamics of the condensate. For shallow-well,
intermediate-well, and deep-well potentials, we find KAM tori and Aubry-Mather
sets to prove that one obtains mostly quasiperiodic dynamics for condensate
wave functions of sufficiently large amplitude, where the minimal amplitude
depends on the experimentally adjustable BEC parameters. We show that this
threshold scales with the square root of the inverse of the two-body scattering
length, whereas the rotation number of tori above this threshold is
proportional to the amplitude. As a consequence, one obtains the same dynamical
picture for lattices of all depths, as an increase in depth essentially only
affects scaling in phase space. Our approach is applicable to periodic
superlattices with an arbitrary number of rationally dependent wave numbers.Comment: 29 pages, 6 figures (several with multiple parts; higher-quality
versions of some of them available at
http://www.its.caltech.edu/~mason/papers), to appear very soon in Journal of
Nonlinear Scienc
Models of spin-orbit coupled oligomers
We address the stability and dynamics of eigenmodes in linearly-shaped
strings (dimers, trimers, tetramers, and pentamers) built of droplets of a
binary Bose-Einstein condensate (BEC). The binary BEC is composed of atoms in
two pseudo-spin states with attractive interactions, dressed by properly
arranged laser fields, which induce the (pseudo-) spin-orbit (SO) coupling. We
demonstrate that the SO-coupling terms help to create eigenmodes of particular
types in the strings. Dimer, trimer, and pentamer eigenmodes of the linear
system, which correspond to the zero eigenvalue (EV, alias chemical potential)
extend into the nonlinear ones, keeping an exact analytical form, while
tetramers do not admit such a continuation, because the respective spectrum
does not contain a zero EV. Stability areas of these modes shrink with the
increasing nonlinearity. Besides these modes, other types of nonlinear states,
which are produced by the continuation of their linear counterparts
corresponding to some nonzero EVs, are found in a numerical form (including
ones for the tetramer system). They are stable in nearly entire existence
regions in trimer and pentamer systems, but only in a very small area for the
tetramers. Similar results are also obtained, but not displayed in detail, for
hexa- and septamers.Comment: Chaos, in pres
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