Transverse Evolution Operator for the Gross-Pitaevskii Equation in Semiclassical Approximation

The Gross-Pitaevskii equation with a local cubic nonlinearity that describes a many-dimensional system in an external field is considered in the framework of the complex WKB-Maslov method. Analytic asymptotic solutions are constructed in semiclassical approximation in a small parameter $\hbar$, $\hbar\to 0$, in the class of functions concentrated in the neighborhood of an unclosed surface associated with the phase curve that describes the evolution of surface vertex. The functions of this class are of the one-soliton form along the direction of the surface normal. The general constructions are illustrated by examples.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Exact Solutions and Symmetry Operators for the Nonlocal Gross-Pitaevskii Equation with Quadratic Potential

The complex WKB-Maslov method is used to consider an approach to the semiclassical integrability of the multidimensional Gross-Pitaevskii equation with an external field and nonlocal nonlinearity previously developed by the authors. Although the WKB-Maslov method is approximate in essence, it leads to exact solution of the Gross-Pitaevskii equation with an external and a nonlocal quadratic potential. For this equation, an exact solution of the Cauchy problem is constructed in the class of trajectory concentrated functions. A nonlinear evolution operator is found in explicit form and symmetry operators (mapping a solution of the equation into another solution) are obtained for the equation under consideration. General constructions are illustrated by examples.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation

We consider the symmetry properties of an integro-differential multidimensional Gross-Pitaevskii equation with a nonlocal nonlinear (cubic) term in the context of symmetry analysis using the formalism of semiclassical asymptotics. This yields a semiclassically reduced nonlocal Gross-Pitaevskii equation, which can be treated as a nearly linear equation, to determine the principal term of the semiclassical asymptotic solution. Our main result is an approach which allows one to construct a class of symmetry operators for the reduced Gross-Pitaevskii equation. These symmetry operators are determined by linear relations including intertwining operators and additional algebraic conditions. The basic ideas are illustrated with a 1D reduced Gross-Pitaevskii equation. The symmetry operators are found explicitly, and the corresponding families of exact solutions are obtained

Symmetry Operators for the Fokker-Plank-Kolmogorov Equation with Nonlocal Quadratic Nonlinearity

The Cauchy problem for the Fokker-Plank-Kolmogorov equation with a nonlocal nonlinear drift term is reduced to a similar problem for the correspondent linear equation. The relation between symmetry operators of the linear and nonlinear Fokker-Plank-Kolmogorov equations is considered. Illustrative examples of the one-dimensional symmetry operators are presented.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Nonlinear Fokker-Planck Equation in the Model of Asset Returns

The Fokker-Planck equation with diffusion coefficient quadratic in space variable, linear drift coefficient, and nonlocal nonlinearity term is considered in the framework of a model of analysis of asset returns at financial markets. For special cases of such a Fokker-Planck equation we describe a construction of exact solution of the Cauchy problem. In the general case, we construct the leading term of the Cauchy problem solution asymptotic in a formal small parameter in semiclassical approximation following the complex WKB-Maslov method in the class of trajectory concentrated functions.Comment: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Symmetry operators of the two-component Gross–Pitaevskii equation with a Manakov-type nonlocal nonlinearity

We consider an integro-differential 2-component multidimensional Gross-Pitaevskii equation with a Manakov-type cubic nonlocal nonlinearity. In the framework of the WKB-Maslov semiclassical formalism, we obtain a semiclassically reduced 2-component nonlocal Gross- Pitaevskii equation determining the leading term of the semiclassical asymptotic solution. For the reduced Gross-Pitaevskii equation we construct symmetry operators which transform arbitrary solution of the equation into another solution. Constructing the symmetry operator is based on the Cauchy problem solution technique and uses an intertwining operator which connects two solutions of the reduced Gross-Pitaevskii equation. General structure of the symmetry operator is illustrated with a 1D case for which a family of symmetry operators is found explicitly and a set of exact solutions is generated

Semiclassical Spectral Series Localized on a Curve for the Gross–Pitaevskii Equation with a Nonlocal Interaction

We propose the approach to constructing semiclassical spectral series for the generalized multidimensional stationary Gross–Pitaevskii equation with a nonlocal interaction term. The eigenvalues and eigenfunctions semiclassically concentrated on a curve are obtained. The curve is described by the dynamic system of moments of solutions to the nonlocal Gross–Pitaevskii equation. We solve the eigenvalue problem for the nonlocal stationary Gross–Pitaevskii equation basing on the semiclassical asymptotics found for the Cauchy problem of the parametric family of linear equations associated with the time-dependent Gross–Pitaevskii equation in the space of extended dimension. The approach proposed uses symmetries of equations in the space of extended dimension

IR spectra of hydrated CaSO4 in the mid-infrared range

Background and Objectives: This work is devoted to the study of the influence of moisture of alabaster (building plaster) samples on the profiles of their IR spectra in the wave number range of 500–4000 cm−1. Materials and Methods: IR spectra of distilled water and alabaster samples with the moisture of 0, 26, 106, 132, 159, 185 and 212% at 23°C were investigated by experimental methods of disturbed total internal reflection. Wave numbers and intensities of components of IR spectra of CaSO4(H2O)n clusters for 0<n<16 were calculated by the methods based on density functional theory with exchange-correlation potential XLYP. Using Gaussian curves with the widths estimated from experiment, the profiles of water valence oscillation bands were determined. When calculating the structure of CaSO4(H2O)n, the positions of atoms in various structural modifications of clusters were optimized. The minimum total energy served as a criterion for choosing the optimal cluster structure, and for the clusters with a large number of atoms, this criterion was applied to an initially selected isomer. Conclusion: On the basis of the calculation results the transformations of the measured spectra (changes of wave numbers and intensities) with changes in the moisture content of the samples have been explained. Comparison of experimental and theoretical spectra in the 3500–3900 cm−1 range allowed to attribute the investigated alabaster powder to a combination of clusters of different sizes:2(CaSO4(H2O)0.5), 2(CaSO4(H2O)0.5 + 0.5H), 4(CaSO4(H2O)0.5), including a cluster of crystalline gypsum: 2(CaSO4(H2O)2). The achieved agreement in the the positions and profiles of the experimental and theoretical water bands in the spectra of samples of different moisture justifies the adequacy of the theoretical description of hydration of CaSO4. &nbsp

Phylogeny, phylogeography and hybridization of Caucasian barbels of the genus Barbus (Actinopterygii, Cyprinidae)

Phylogenetic relationships and phylogeography of six species of Caucasian barbels, the genus Barbus s. str., were studied based on extended geographic coverage and using mtDNA and nDNA markers. Based on 27 species studied, matrilineal phylogeny of the genus Barbus is composed of two clades ¿ (a) West European clade, (b) Central and East European clade. The latter comprises two subclades: (b1) Balkanian subclade, and (b2) Ponto-Caspian one that includes 11 lineages mainly from Black and Caspian Sea drainages. Caucasian barbels are not monophyletic and subdivided for two groups. The Black Sea group encompasses species from tributaries of Black Sea including re-erected B. rionicus and excluding B. kubanicus. The Caspian group includes B. ciscaucasicus, B. cyri (with B. goktschaicus that might be synonymized with B. cyri), B. lacerta from the Tigris-Euphrates basin and B. kubanicus from the Kuban basin. Genetic structure of Black Sea barbels was influenced by glaciation-deglaciation periods accompanying by freshwater phases, periods of migration and colonization of Black Sea tributaries. Intra- and intergeneric hybridization among Caucasian barbines was revealed. In the present study, we report about finding of B. tauricus in the Kuban basin, where only B. kubanicus was thought to inhabit. Hybrids between these species were detected based on both mtDNA and nDNA markers. Remarkably, Kuban population of B. tauricus is distant to closely located conspecific populations and we consider it as relic. We highlight revealing the intergeneric hybridization between evolutionary tetraploid (2n=100) B. goktschaicus and evolutionary hexaploid (2n=150) Capoeta sevangi in Lake Sevan.The study was supported by Russian Science Foundation (grant no. 15-14-10020); final stage of the study was supported by Russian Foundation for Basic Research (grants nos. 18-54-05003 and 19-04-00719)