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

GROUP ANALYSIS AND SOME EXACT SOLUTIONS FOR THE THERMAL BOUNDARY LAYER

We perform the group analysis of the thermal boundary layer in laminar flow. We obtain the classification of the solutions in terms of the asymptotic velocity. Some solutions of the boundary layer equations, for some distributions of outer flow velocity, are obtained also

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

Singularity analysis and integrability for a HNLS equation governing pulse propagation in a generic fiber optics

Taking into account many developments in fiber optics communications, we propose a higher nonlinear Schrödinger equation (HNLS) with variable coefficients, more general than that in [R. Essiambre, G.P. Agrawal, Opt. Commun. 131 (1996) 274], which governs the propagation of ultrashort pulses in a fiber optics with generic variable dispersion. The study of this equation is performed using the Painlevé test and the zero-curvature method. Also, we prove the equivalence between this equation and its anomalous integrable counterpart (the so-called Sasa-Satsuma equation). Finally, in view of its physical relevance, we present a soliton solution which represents the propagation of ultrashort pulses in a dispersion decreasing fiber