275 research outputs found
Unstaggered-staggered solitons on one- and two-dimensional two-component discrete nonlinear Schr\"{o}dinger lattices
We study coupled unstaggered-staggered soliton pairs emergent from a system
of two coupled discrete nonlinear Schr\"{o}dinger (DNLS) equations with the
self-attractive on-site self-phase-modulation nonlinearity, coupled by the
repulsive cross-phase-modulation interaction, on 1D and 2D lattice domains.
These mixed modes are of a "symbiotic" type, as each component in isolation may
only carry ordinary unstaggered solitons. While most work on DNLS systems
addressed symmetric on-site-centered fundamental solitons, these models give
rise to a variety of other excited states, which may also be stable. The
simplest among them are antisymmetric states in the form of discrete twisted
solitons, which have no counterparts in the continuum limit. In the extension
to 2D lattice domains, a natural counterpart of the twisted states are vortical
solitons. We first introduce a variational approximation (VA) for the solitons,
and then correct it numerically to construct exact stationary solutions, which
are then used as initial conditions for simulations to check if the stationary
states persist under time evolution. Two-component solutions obtained include
(i) 1D fundamental-twisted and twisted-twisted soliton pairs, (ii) 2D
fundamental-fundamental soliton pairs, and (iii) 2D vortical-vortical soliton
pairs. We also highlight a variety of other transient dynamical regimes, such
as breathers and amplitude death. The findings apply to modeling binary
Bose-Einstein condensates, loaded in a deep lattice potential, with identical
or different atomic masses of the two components, and arrays of bimodal optical
waveguides.Comment: to be published in Communications in Nonlinear Science and Numerical
Simulatio
Description of hard sphere crystals and crystal-fluid interfaces: a critical comparison between density functional approaches and a phase field crystal model
In materials science the phase field crystal approach has become popular to
model crystallization processes. Phase field crystal models are in essence
Landau-Ginzburg-type models, which should be derivable from the underlying
microscopic description of the system in question. We present a study on
classical density functional theory in three stages of approximation leading to
a specific phase field crystal model, and we discuss the limits of
applicability of the models that result from these approximations. As a test
system we have chosen the three--dimensional suspension of monodisperse hard
spheres. The levels of density functional theory that we discuss are
fundamental measure theory, a second-order Taylor expansion thereof, and a
minimal phase-field crystal model. We have computed coexistence densities,
vacancy concentrations in the crystalline phase, interfacial tensions and
interfacial order parameter profiles, and we compare these quantities to
simulation results. We also suggest a procedure to fit the free parameters of
the phase field crystal model.Comment: 21 page
Landau-Ginzburg Description of Boundary Critical Phenomena in Two Dimensions
The Virasoro minimal models with boundary are described in the
Landau-Ginzburg theory by introducing a boundary potential, function of the
boundary field value. The ground state field configurations become non-trivial
and are found to obey the soliton equations. The conformal invariant boundary
conditions are characterized by the reparametrization-invariant data of the
boundary potential, that are the number and degeneracies of the stationary
points. The boundary renormalization group flows are obtained by varying the
boundary potential while keeping the bulk critical: they satisfy new selection
rules and correspond to real deformations of the Arnold simple singularities of
A_k type. The description of conformal boundary conditions in terms of boundary
potential and associated ground state solitons is extended to the N=2
supersymmetric case, finding agreement with the analysis of A-type boundaries
by Hori, Iqbal and Vafa.Comment: 42 pages, 13 figure
Analytical Proof of Space-Time Chaos in Ginzburg-Landau Equations
We prove that the attractor of the 1D quintic complex Ginzburg-Landau
equation with a broken phase symmetry has strictly positive space-time entropy
for an open set of parameter values. The result is obtained by studying chaotic
oscillations in grids of weakly interacting solitons in a class of
Ginzburg-Landau type equations. We provide an analytic proof for the existence
of two-soliton configurations with chaotic temporal behavior, and construct
solutions which are closed to a grid of such chaotic soliton pairs, with every
pair in the grid well spatially separated from the neighboring ones for all
time. The temporal evolution of the well-separated multi-soliton structures is
described by a weakly coupled lattice dynamical system (LDS) for the
coordinates and phases of the solitons. We develop a version of normal
hyperbolicity theory for the weakly coupled LDSs with continuous time and
establish for them the existence of space-time chaotic patterns similar to the
Sinai-Bunimovich chaos in discrete-time LDSs. While the LDS part of the theory
may be of independent interest, the main difficulty addressed in the paper
concerns with lifting the space-time chaotic solutions of the LDS back to the
initial PDE. The equations we consider here are space-time autonomous, i.e. we
impose no spatial or temporal modulation which could prevent the individual
solitons in the grid from drifting towards each other and destroying the
well-separated grid structure in a finite time. We however manage to show that
the set of space-time chaotic solutions for which the random soliton drift is
arrested is large enough, so the corresponding space-time entropy is strictly
positive
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