Exact Solutions and hypothesis on Phase Transition in the Polyelectrolite Model of DNA

The two-dimensional generalization of the Polyelectolite model of DNA is proposed. It is reduced to the boundary problem for nonlinear completely integrable equation $\sh$-Gordon. In the linearisible version the exact solution is constructed and its asymptotic is found. The soliton solutions of nonlinear equation are calculated that allowed tells about the possibility of the structural phase transition in the considered system (DNA+polyelectrolite) on the temperature.Comment: Latex, 12 page

Darboux Transformation and Exact Solutions in the model of Cylindrically Symmetrical Chiral Field

The application of the Darboux Transformation method to the integrable model of Cylindrically Symmetrical Chiral field has been considered. The associated linear system of matrix equations has been proposed and the properties of symmetrie for its solutions has been obtained. The necessary form of Darboux Transformation has been found and formal one- and N-soliton solutions have been constructed. With the use of Pohlmayer's Transformation the equation sin-Gordon type have been given and the hypothesis about its integrability has been deduced.Comment: 10 pages. The talk at the International Seminar "Day on Diffraction". June 1-3, 1999. St.Petersburg, Russi

Darboux Transformation for the Non-stationary Shr\"odinger Equation

The Lax representation for the nonstationary Schr\"odinger equation with rather arbitrary potential is proposed. Some examples of the construction of exact solutions are given by means of Darboux Transformation method

Nonlinear Sigma Model, Zakharov-Shabat Method, and New Exact Forms of the Minimal Surfaces in R3R^3

General formulas for the construction of exact solutions of the equation of the minimal surface in $R^3$, which appears in various physical problems, have been derived by the Zakharov-Shabat "dressing" method. Particular examples are considered.Comment: 8 page

Two approaches for Helmholtz equation: generalized Darboux Trasformation and the method of d-bar problem

Two approaches to solution of the two-dimensional Helmholtz equation with a "wave number" are proposed. The results can be applied both in numerical areas of physics and in the theory of nonlinear equations. The first approach is based on the requirement of the covariance of equation under the generalized Darboux transformation (Moutard transformation). It allows to construct a new solution of equation, using a given initial solution of the equation. Simultaneously we obtain the "dressing" relation for the "wave number". The simplest examples of the approach are considered in detail. In the second approach the Green-Cauchy formula (the $\bar \partial$-method) is applied to reduce the solution of the equation to the solution of a system of singular integral equations.Comment: 11 pages. In: Proceedings of the International Seminar "Day on Diffraction- 2003", St.Petersburg, Russia, June 24-27, 2003, pp.60-7

Hydrodynamical vortex on the plane

The detailed analysis of model of the hydrodynamical vortice on a plane is executed. The derivation of the corresponding equation and its simplified variant is given, a partial solutions are constructed. The question on application of Darboux transformation is considered.Comment: 11 page

Spiral-Logarithmic Structure in a Heisenberg Ferromagnet

Spiral-logarithmic structure is suggested as a stationary solution of a modified equation for the Heisenberg model, and the single- and N-soliton solutions are constructed on this base.Comment: LaTeX, 5 pages, no figure

Some notes on Ishimori's magnet model

Gauge transformation properties for an associated linear system of model Ishimori's magnet model have been discussed. Explicit formulas for the gauge transformation matrix have been obtained. Darboux Transformation has been suggested and appropriate dressing relations have been found.Comment: 11 page

Orienting Asymmetric Molecules by Laser Fields with Skewed Polarization

We study interaction of generic asymmetric molecules with a pair of strong time-delayed short laser pulses with crossed linear polarizations. We show that such an excitation not only provides unidirectional rotation of the most polarizable molecular axis, but also induces a directed torque along this axis, which results in the transient orientation of the molecules. The asymmetric molecules are chiral in nature and different molecular enantiomers experience the orienting action in opposite directions causing out-of-phase oscillation of their dipole moments. The resulting microwave radiation was recently suggested to be used for analysis/discrimination of chiral molecular mixtures. We reveal the mechanism behind this laser induced orientation effect, show that it is classical in nature, and envision further applications of light with skewed polarization.Comment: 8 pages, 5 figure

Acoustical Properties of Superfluid Helium in Confined Geometry

The problem studied in this paper is to obtain the equations describing sound propagation in a consolidated porous medium filled with superfluid, determine the elastic coefficients, appearing in the equations, in terms of physically measurable quantities, and calculate the propagation velocities of transverse and longitudinal waves at high and low oscillating frequencies. In general, the obtained equations describe all volume modes that can propagate in a porous medium saturated with superfluid for any values of the porosity and frequencies. The derived equations are applied to the most important particular case when the normal component of superfluid helium is locked inside a highly porous media (aerogel, Im-helium sample) by viscous forces. For this case the velocities of two longitudinal sound modes and transverse mode are calculated from the derived equations. There are established the coupling between temperature and pressure oscillations in these fast and slow modes.Comment: 2