The Fourier method for the linearized Davey-Stewartson I equation

The linearized Davey-Stewartson equation with varing coefficients is solved by Fourier method. The approach uses the inverse scattering transform for the Davey-Stewartson equation.Comment: 4 pages, LaTe

Emission of autoresonant trajectories and thresholds of resonant pumping

We study an autoresonant asymptotic behaviour for nonlinear oscillators under slowly changing frequency and amplitude of external driver. As a result we obtain formulas for threshold values of amplitude and frequency of the driver when autoresonant behaviour for the nonlinear oscillator is observed. Also we study a capture into resonance and emission out of the resonance for trajectories of the oscillator. A measure of autoresonant asymptotic behaviours for nonlinear oscillator is obtained.Comment: 15 pages, 6 figure

Asymptotics of soliton solution for the perturbed Davey-Stewartson-1 equations

The dromion of the Davey-Stewartson-1 equation is studied under perturbation on the large time.Comment: Mistake and misprints are corrected. Latex, 14 page

Direct Photons in Ion Collisions at FAIR Energies

Estimations of prompt photon production at FAIR (Facility for Antiproton and Ion Research) energies using the extrapolation of existing data are presented. About $10^{-4}$ prompt $\gamma$ with $p_{t}>$2 GeV/c per Au+Au central event at 25 AGeV are expected. With the planed beam intensity $10^{9}/s$, 1% interaction rate and 10% centrality, at CBM (Compressed Baryonic Matter) experiment one can expect prompt $\gamma$ rate ~100/s. Predictions for direct photons by some generators (PYTHIA, UrQMD, RQMD, HSD, HIJING) are analyzed. One of the main sources of direct photons (due to meson scatterings $\pi\rho\to\pi\gamma, \pi\pi\to\rho\gamma$) is not implemented in the heavy-ion generators. Corresponding cross-sections for this source have been prepared for implementation into the HSD code. Main experimental methods to study direct photons (subtraction method, momentum correlations method and internal conversion method) are shortly reviewed. High intensity beam, good tracking and good $e^{\pm}$ particle identification of the CBM detector favor to measure direct photons by all the methods.Comment: 7 pages,5 figures,Talk at Baldin ISHEPP XVIII, Dubna, 25 - 30 September 200

Asymptotic approach for the rigid condition of appearance of the oscillations in the solution of the Painleve-2 equation

The asymptotic solution for the Painleve-2 equation with small parameter is considered. The solution has algebraic behavior before point $t_*$ and fast oscillating behavior after the point $t_*$. In the transition layer the behavior of the asymptotic solution is more complicated. The leading term of the asymptotics satisfies the Painleve-1 equation and some elliptic equation with constant coefficients, where the solution of the Painleve-1 equation has poles. The uniform smooth asymptotics are constructed in the interval, containing the critical point $t_*$.Comment: Latex, 18 page

Absolutely continuous spectrum of Stark operators

We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely continuous spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is $|q(x)| \leq C(1+|x|)^{-{1/4}-\epsilon};$ in the smoothness direction, a sufficient condition in H\"older classes is $q \in C^{{1/2}+\epsilon}(\reals)$. On the other hand, we show that there exist potentials which both satisfy $|q(x)| \leq C(1+|x|)^{-1/4}$ and belong to $C^{1/2}(\reals)$ for which the spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.Comment: 29 page

Tunable RKKY interaction in a double quantum dot nanoelectromechanical device

We propose a realization of mechanically tunable Ruderman-Kittel-Kasuya-Yosida interaction in a double quantum dot nanoelectromechanical device. The coupling between spins of two quantum dots suspended above a metallic plate is mediated by conduction electrons. We show that the spin-mechanical interaction can be driven by a slow modulation of charge density in the metallic plate. We propose to use Stuckelberg oscillations as a sensitive tool for detection of the spin and charge states of the coupled quantum dots. Theory of mechanical back action induced by a dynamical spin-spin interaction is discussed

The slowly passage through the resonances and wave packets with the different carriers

Solution of the nonlinear Klein-Gordon equation perturbed by small external force is investigated. The perturbation is represented by finite collections of harmonics. The frequencies of the perturbation vary slowly and pass through the resonant values consecutively. The resonances lead to the sequence of the wave packets with the different fast oscillated carriers. Full asymptotic description of this process is presented.Comment: 24 pages, LaTe

Scattering of solitons on resonance

We investigate a propagation of solitons for nonlinear Schrodinger equation under small driving force. The driving force passes the resonance. The process of scattering on the resonance leads to changing of number of solitons. After the resonance the number of solitons depends on the amplitude of the driving force.Comment: LaTeX, 10 page

The solution of the Painleve equations as special functions of catastrophes, defined by a rejection in these equations of terms with derivative

The relation between the Painleve equations and the algebraic equations with the catastrophe theory point of view are considered. The asymptotic solutions with respect to the small parameter of the Painleve equations different types are discussed. The qualitative analysis of the relation between algebraic and fast oscillating solutions is done for Painleve-2 as an example.Comment: Latex, 15 page