76 research outputs found
Semiclassical dynamics of quasi-one-dimensional, attractive Bose-Einstein condensates
The strongly interacting regime for attractive Bose-Einstein condensates
(BECs) tightly confined in an extended cylindrical trap is studied. For
appropriately prepared, non-collapsing BECs, the ensuing dynamics are found to
be governed by the one-dimensional focusing Nonlinear Schr\"odinger equation
(NLS) in the semiclassical (small dispersion) regime. In spite of the
modulational instability of this regime, some mathematically rigorous results
on the strong asymptotics of the semiclassical limiting solutions were obtained
recently. Using these results, "implosion-like" and "explosion-like" events are
predicted whereby an initial hump focuses into a sharp spike which then expands
into rapid oscillations. Seemingly related behavior has been observed in
three-dimensional experiments and models, where a BEC with a sufficient number
of atoms undergoes collapse. The dynamical regimes studied here, however, are
not predicted to undergo collapse. Instead, distinct, ordered structures,
appearing after the "implosion", yield interesting new observables that may be
experimentally accessible.Comment: 9 pages, 3 figure
Universality in the profile of the semiclassical limit solutions to the focusing Nonlinear Schroedinger equation at the first breaking curve
We consider the semiclassical (zero-dispersion) limit of the one-dimensional
focusing Nonlinear Schroedinger equation (NLS) with decaying potentials. If a
potential is a simple rapidly oscillating wave (the period has the order of the
semiclassical parameter epsilon) with modulated amplitude and phase, the
space-time plane subdivides into regions of qualitatively different behavior,
with the boundary between them consisting typically of collection of piecewise
smooth arcs (breaking curve(s)). In the first region the evolution of the
potential is ruled by modulation equations (Whitham equations), but for every
value of the space variable x there is a moment of transition (breaking), where
the solution develops fast, quasi-periodic behavior, i.e., the amplitude
becomes also fastly oscillating at scales of order epsilon. The very first
point of such transition is called the point of gradient catastrophe. We study
the detailed asymptotic behavior of the left and right edges of the interface
between these two regions at any time after the gradient catastrophe. The main
finding is that the first oscillations in the amplitude are of nonzero
asymptotic size even as epsilon tends to zero, and they display two separate
natural scales; of order epsilon in the parallel direction to the breaking
curve in the (x,t)-plane, and of order epsilon ln(epsilon) in a transversal
direction. The study is based upon the inverse-scattering method and the
nonlinear steepest descent method.Comment: 40 pages, 10 figure
Dam break problem for the focusing nonlinear Schr\"odinger equation and the generation of rogue waves
We propose a novel, analytically tractable, scenario of the rogue wave
formation in the framework of the small-dispersion focusing nonlinear
Schr\"odinger (NLS) equation with the initial condition in the form of a
rectangular barrier (a "box"). We use the Whitham modulation theory combined
with the nonlinear steepest descent for the semi-classical inverse scattering
transform, to describe the evolution and interaction of two counter-propagating
nonlinear wave trains --- the dispersive dam break flows --- generated in the
NLS box problem. We show that the interaction dynamics results in the emergence
of modulated large-amplitude quasi-periodic breather lattices whose amplitude
profiles are closely approximated by the Akhmediev and Peregrine breathers
within certain space-time domain. Our semi-classical analytical results are
shown to be in excellent agreement with the results of direct numerical
simulations of the small-dispersion focusing NLS equation.Comment: 29 pages, 15 figures, major revisio
Statistics of extreme events in integrable turbulence
We use the spectral kinetic theory of soliton gas to investigate the
likelihood of extreme events in integrable turbulence described by the
one-dimensional focusing nonlinear Schr\"odinger equation (fNLSE). This is done
by invoking a stochastic interpretation of the inverse scattering transform for
fNLSE and analytically evaluating the kurtosis of the emerging random nonlinear
wave field in terms of the spectral density of states of the corresponding
soliton gas. We then apply the general result to two fundamental scenarios of
the generation of integrable turbulence: (i) the asymptotic development of the
spontaneous (noise induced) modulational instability of a plane wave, and (ii)
the long-time evolution of strongly nonlinear, partially coherent waves. In
both cases, involving the bound state soliton gas dynamics, the analytically
obtained values of the kurtosis are in perfect agreement with those inferred
from direct numerical simulations of the the fNLSE, providing the long-awaited
theoretical explanation of the respective rogue wave statistics. Additionally,
the evolution of a particular non-bound state gas is considered providing
important insights related to the validity of the so-called virial theorem.Comment: 11 pages, 5 figure
Meromorphic differentials with imaginary periods on degenerating hyperelliptic curves
We provide a direct and explicit proof that imaginary (real) normalized differentials of the second kind with prescribed polar part do not develop additional singularities as the underlying hyperelliptic Riemann surface degenerates in an arbitrary way
A HYBRID DIFFERENTIAL EVOLUTION FOR NON-SMOOTH OPTIMIZATION PROBLEMS
Solving high dimentional, multimodal, non-smooth global optimization problems faces challenges concerning quality of solution, computational costs or even the impossibility of solving the problem. Evolutionary algorithms, in particular, differential evolution algorithm proved itself as good method of global optimization. On the other side, approach based on subgradient methods are good for optimizing non-smooth functions. Combination of these two approaches enables to improve the quality of the algorithm, using the best features of both methods. In this paper, a new hybrid evolutionary approach based on differential evolution and subgradient algorithm as the local search procedure is proposed. Behavior of the proposed SSGDE algorithm was studied in a numerical experiment on three groups of generated tests. Comparison of the new hybrid algorithm with the pure DE approach showed the advantage of the SSGDE. It has been experimentally established that the proposed method finds the global minimum in the best way for all considered dimensions of the problem with respect to the differential evolution method. The SSGDE algorithm showed the best results with a significant increase in the number of functions
Translationally invariant nonlinear Schrodinger lattices
Persistence of stationary and traveling single-humped localized solutions in
the spatial discretizations of the nonlinear Schrodinger (NLS) equation is
addressed. The discrete NLS equation with the most general cubic polynomial
function is considered. Constraints on the nonlinear function are found from
the condition that the second-order difference equation for stationary
solutions can be reduced to the first-order difference map. The discrete NLS
equation with such an exceptional nonlinear function is shown to have a
conserved momentum but admits no standard Hamiltonian structure. It is proved
that the reduction to the first-order difference map gives a sufficient
condition for existence of translationally invariant single-humped stationary
solutions and a necessary condition for existence of single-humped traveling
solutions. Other constraints on the nonlinear function are found from the
condition that the differential advance-delay equation for traveling solutions
admits a reduction to an integrable normal form given by a third-order
differential equation. This reduction also gives a necessary condition for
existence of single-humped traveling solutions. The nonlinear function which
admits both reductions defines a two-parameter family of discrete NLS equations
which generalizes the integrable Ablowitz--Ladik lattice.Comment: 24 pages, 4 figure
Travelling kinks in discrete phi^4 models
In recent years, three exceptional discretizations of the phi^4 theory have
been discovered [J.M. Speight and R.S. Ward, Nonlinearity 7, 475 (1994); C.M.
Bender and A. Tovbis, J. Math. Phys. 38, 3700 (1997); P.G. Kevrekidis, Physica
D 183, 68 (2003)] which support translationally invariant kinks, i.e. families
of stationary kinks centred at arbitrary points between the lattice sites. It
has been suggested that the translationally invariant stationary kinks may
persist as 'sliding kinks', i.e. discrete kinks travelling at nonzero
velocities without experiencing any radiation damping. The purpose of this
study is to check whether this is indeed the case. By computing the Stokes
constants in beyond-all-order asymptotic expansions, we prove that the three
exceptional discretizations do not support sliding kinks for most values of the
velocity - just like the standard, one-site, discretization. There are,
however, isolated values of velocity for which radiationless kink propagation
becomes possible. There is one such value for the discretization of Speight and
Ward and three 'sliding velocities' for the model of Kevrekedis.Comment: To be published in Nonlinearity. 22 pages, 5 figures. Extensive
clarifications to the text have been mad
Experimental and theoretical study of the acylation reaction of aminopyrazoles with aryl and methoxymethyl substituents
As a result of the chain of transformations from 1,3-butanedione with aryl and methoxy sub-stituents through nitrosation and cyclization with hydrazine, the corresponding nitrosopyrazoles and aminopyrazoles were synthesized. According to this scheme, eight new previously unknown com-pounds were obtained. Their structures were established by the methods of IR, UV, 1H NMR, 13C NMR spectroscopy and mass spectrometry. DFT method of quantum-chemical calculations showed that obtained aminopyrazoles can exist as two tautomers; it was also confirmed by NMR 1H spec-troscopy data. In the case of acylation, an isomer is formed, where aryl substituent takes place in the fifth, rather than in the third position of the pyrazole ring, as shown by the DFT calculations
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