Volume 9, Number 1
Volume 9, Number 1, 2004
Troubetzkoy S.
Abstract
We show that almost all billiard trajectories return parallel to themselves for rank $1$, ergodic polygons. Applications are given to the existence of periodic trajectories.

Tamizhmani K. M., Grammaticos B., Carstea A. S., Ramani A.
Abstract
We present a detailed study of the properties of two qdiscrete Painlevé IV equations: singularity structure, bilinear forms, autoBäcklund/Schlesinger transformations, rational solutions as well as special solutions obtained through second or firstorder linear equations.

Kistovich A. V., Chashechkin Y. D.
Abstract
The theoretical study of steady vortex motion of homogeneous ideal heavy fluid with a free surface by methods of differential geometry is presented. The main idea of methods is based on suggestion that a velocity field is formed by geodesic flows at some surfaces. For steady flow integral flow lines are geodesics on the second order surfaces being parameterized and located in the space occupied by the fluid. In this case both Euler and continuity equations are transformed into equations for inner geometry parameters. Conditions on external parameters are derived from boundary conditions of the problem. The investigation of properties of generalized Rankine vortex that is vertical vortex flow contacting with a free surface is done. In supplement to the classical Rankine vortex these vortices are characterized by all finite specific integral invariants. The constructed set of explicit solutions depend on a unique parameter, which can be defined experimentally through measurements of depth and shape of a near surface hole produced by the vortex.

Kozlov V. V.
Abstract
A collisionless continuous medium in Euclidean space is discussed, i.e. a continuum of free particles moving inertially, without interacting with each other. It is shown that the distribution density of such medium is weakly converging to zero as time increases indefinitely. In the case of Maxwell's velocity distribution of particles, this density satisfies the wellknown diffusion equation, the diffusion coefficient increasing linearly with time.

AitMokhtar S.
Abstract
We consider a class of third order homogenous differential equation whose coefficients are analytic functions of the complex variable. We find necessary and sufficient conditions about the coefficients in order that the only movable singularities of the solutions may be poles.

Kovalev A. M.
Abstract
The problem on inclusion of an invariant manifold of a dynamical system into a set of integral manifolds is considered. It is shown that this inclusion is always possible unless the invariant manifold is a singular one (consisting of singular points of the system) of codimension one. It enables us to study invariant manifolds using the equation for integrals instead of the LeviCivita equations containing uncertain factors. The defining role of the singular manifolds in shaping a phase portrait of a dynamical system is established and the following from this integrals properties are obtained. The results are applied to the analysis of solutions of the Euler–Poisson equations. A new treatment of the fourth integrals in Euler's, Lagrange's, Kovalevskaya's cases is proposed. It is proved that under the Hess conditions there is a fourth integral of which special cases are Euler's and Lagrange's integrals as well as the Hess and the Dokshevich solutions.
