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

Gradient catastrophe and flutter in vortex filament dynamics

Gradient catastrophe and flutter instability in the motion of vortex filament within the localized induction approximation are analyzed. It is shown that the origin if this phenomenon is in the gradient catastrophe for the dispersionless Da Rios system which describes motion of filament with slow varying curvature and torsion. Geometrically this catastrophe manifests as a rapid oscillation of a filament curve in a point that resembles the flutter of airfoils. Analytically it is the elliptic umbilic singularity in the terminology of the catastrophe theory. It is demonstrated that its double scaling regularization is governed by the Painlev\'e-I equation.Comment: 11 pages, 3 figures, typos corrected, references adde

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

On critical behaviour in systems of Hamiltonian partial differential equations

We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\ue9-I (PI) equation or its fourth-order analogue P2I. As concrete examples, we discuss nonlinear Schr\uf6dinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture

Modern Trends of Organic Chemistry in Russian Universities

© 2018, Pleiades Publishing, Ltd. This review is devoted to the scientific achievements of the departments of organic chemistry in higher schools of Russia within the past decade

Formation of Fuzzy Patterns in Logical Analysis of Data Using a Multi-Criteria Genetic Algorithm

The formation of patterns is one of the main stages in logical data analysis. Fuzzy approaches to pattern generation in logical analysis of data allow the pattern to cover not only objects of the target class, but also a certain proportion of objects of the opposite class. In this case, pattern search is an optimization problem with the maximum coverage of the target class as an objective function, and some allowed coverage of the opposite class as a constraint. We propose a more flexible and symmetric optimization model which does not impose a strict restriction on the pattern coverage of the opposite class observations. Instead, our model converts such a restriction (purity restriction) into an additional criterion. Both, coverage of the target class and the opposite class are two objective functions of the optimization problem. The search for a balance of these criteria is the essence of the proposed optimization method. We propose a modified evolutionary algorithm based on the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) to solve this problem. The new algorithm uses pattern formation as an approximation of the Pareto set and considers the solution’s representation in logical analysis of data and the informativeness of patterns. We have tested our approach on two applied medical problems of classification under conditions of sample asymmetry: one class significantly dominated the other. The classification results were comparable and, in some cases, better than the results of commonly used machine learning algorithms in terms of accuracy, without losing the interpretability