PAINLEVE' ANALYSIS AND EXACT SOLUTIONS FOR THE COUPLET BURGERS SYSTEM

We perform the Painlev\ue8 test to a system of two coupled Burgers-type equations which fails to satisfy the Painlev\ue8 test. In order to obtain a class of solutions, we use a slightly modified version of the test. These solutions are expressed in terms of the Airy functions. We also give the travelling wave solutions,expressed in terms of the trigonometric and hyperbolic functions

On exact solutions for quantum particles with spin S= 0, 1/2, 1 and de Sitter event horizon

Exact wave solutions for particles with spin 0, 1/2 and 1 in the static coordinates of the de Sitter space-time model are examined in detail. Firstly, for a scalar particle, two pairs of linearly independent solutions are specified explicitly: running and standing waves. A known algorithm for calculation of the reflection coefficient $R_{\epsilon j}$ on the background of the de Sitter space-time model is analyzed. It is shown that the determination of R_{\epsilon j} requires an additional constrain on quantum numbers \epsilon \rho / \hbar c >> j, where \rho is a curvature radius. When taken into account of this condition, the R_{\epsilon j} vanishes identically. It is claimed that the calculation of the reflection coefficient R_{\epsilon j} is not required at all because there is no barrier in an effective potential curve on the background of the de Sitter space-time. The same conclusion holds for arbitrary particles with higher spins, it is demonstrated explicitly with the help of exact solutions for electromagnetic and Dirac fields.Comment: 30 pages. This paper is an updated and more comprehensive version of the old paper V.M. Red'kov. On Particle penetrating through de Sitter horizon. Minsk (1991) 22 pages Deposited in VINITI 30.09.91, 3842 - B9

Exact Solutions of the Two Dimensional Boussinesq and Dispersive Water Waves Equations

In this paper two-dimensional Boussinesq and dispersive water waves equations are investigated in exact solutions. The Exp-function method is used for seeking exact solutions of the equations through symbolic computation