198 research outputs found
Dark solitons in external potentials
We consider the persistence and stability of dark solitons in the
Gross-Pitaevskii (GP) equation with a small decaying potential. We show that
families of black solitons with zero speed originate from extremal points of an
appropriately defined effective potential and persist for sufficiently small
strength of the potential. We prove that families at the maximum points are
generally unstable with exactly one real positive eigenvalue, while families at
the minimum points are generally unstable with exactly two complex-conjugated
eigenvalues with positive real part. This mechanism of destabilization of the
black soliton is confirmed in numerical approximations of eigenvalues of the
linearized GP equation and full numerical simulations of the nonlinear GP
equation with cubic nonlinearity. We illustrate the monotonic instability
associated with the real eigenvalues and the oscillatory instability associated
with the complex eigenvalues and compare the numerical results of evolution of
a dark soliton with the predictions of Newton's particle law for its position.Comment: 39 pages, 10 figure
Exploring Critical Points of Energy Landscapes: From Low-Dimensional Examples to Phase Field Crystal PDEs
In the present work we explore the application of a few root-finding methods
to a series of prototypical examples. The methods we consider include: (a) the
so-called continuous-time Nesterov (CTN) flow method; (b) a variant thereof
referred to as the squared-operator method (SOM); and (c) the the joint action
of each of the above two methods with the so-called deflation method. More
traditional methods such as Newton's method (and its variant with deflation)
are also brought to bear. Our toy examples start with a naive one
degree-of-freedom (dof) system to provide the lay of the land. Subsequently, we
turn to a 2-dof system that is motivated by the reduction of an
infinite-dimensional, phase field crystal (PFC) model of soft matter
crystallisation. Once the landscape of the 2-dof system has been elucidated, we
turn to the full PDE model and illustrate how the insights of the
low-dimensional examples lead to novel solutions at the PDE level that are of
relevance and interest to the full framework of soft matter crystallization.Comment: 17 pages, 16 figure
Motion of discrete solitons assisted by nonlinearity management
We demonstrate that periodic modulation of the nonlinearity coefficient in
the discrete nonlinear Schr\"{o}dinger (DNLS) equation can strongly facilitate
creation of traveling solitons in the lattice. We predict this possibility in
an analytical form, and test it in direct simulations. Systematic simulations
reveal several generic dynamical regimes, depending on the amplitude and
frequency of the time modulation, and on initial thrust which sets the soliton
in motion. These regimes include irregular motion, regular motion of a decaying
soliton, and regular motion of a stable one. The motion may occur in both the
straight and reverse directions, relative to the initial thrust. In the case of
stable motion, extremely long simulations in a lattice with periodic boundary
conditions demonstrate that the soliton keeps moving as long as we can monitor
without any visible loss. Velocities of moving stable solitons are in good
agreement with the analytical prediction, which is based on requiring a
resonance between the ac drive and motion of the soliton through the periodic
potential. All the generic dynamical regimes are mapped in the model's
parameter space. Collisions between moving stable solitons are briefly
investigated too, with a conclusion that two different outcomes are possible:
elastic bounce, or bounce with mass transfer from one soliton to the other. The
model can be realized experimentally in a Bose-Einstein condensate trapped in a
deep optical lattice
Discrete breathers in Φ4 and related models
In this Chapter, we touch upon the wide topic of discrete breather (DB)
formation with a special emphasis on the prototypical system of interest, namely
the 4 model. We start by introducing the model and discussing some of the application
areas/motivational aspects of exploring time periodic, spatially localized
structures, such as the DBs. Our main emphasis is on the existence, and especially
on the stability features of such solutions.We explore their spectral stability numerically,
as well as in special limits (such as the vicinity of the so-called anti-continuum
limit of vanishing coupling) analytically. We also provide and explore a simple, yet
powerful stability criterion involving the sign of the derivative of the energy vs.
frequency dependence of such solutions. We then turn our attention to nonlinear
stability, bringing forth the importance of a topological notion, namely the Krein
signature. Furthermore, we briefly touch upon linearly and nonlinearly unstable dynamics
of such states. Some special aspects/extensions of such structures are only
touched upon, including moving breathers and dissipative variations of the model
and some possibilities for future work are highlighted. While this Chapter by no
means aspires to be comprehensive, we hope that it provides some recent developments
(a large fraction of which is not included in time-honored DB reviews) and
associated future possibilities.AEI/FEDER, (UE) MAT2016- 79866-
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