Explicit form of the Yablonskii - Vorob'ev polynomials

Special polynomials associated with rational solutions of the second Painlev\'{e} equation and other members of its hierarchy are discussed. New approach, which allows one to construct each polynomial is presented. The structure of the polynomials is established. Formulas of their coefficients are found. Correlations between the roots of every polynomial are obtained.Comment: 21 page

Newton polygons for finding exact solutions

A method for finding exact solutions of nonlinear differential equations is presented. Our method is based on the application of the Newton polygons corresponding to nonlinear differential equations. It allows one to express exact solutions of the equation studied through solutions of another equation using properties of the basic equation itself. The ideas of power geometry are used and developed. Our approach has a pictorial rendition, which is is illustrative and effective. The method can be also applied for finding transformations between solutions of the differential equations. To demonstrate the method application exact solutions of several equations are found. These equations are: the Korteveg - de Vries - Burgers equation, the generalized Kuramoto - Sivashinsky equation, the fourth - order nonlinear evolution equation, the fifth - order Korteveg - de Vries equation, the modified Korteveg - de Vries equation of the fifth order and nonlinear evolution equation of the sixth order for the turbulence description. Some new exact solutions of nonlinear evolution equations are given.Comment: 24 pages, 10 figure

Multi-particle dynamical systems and polynomials

Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi--particle dynamical system by finding polynomial solutions of a partial differential equations is described. The method enables one to integrate a wide class of polynomial multi--particle dynamical systems. The general solutions of certain dynamical systems related to linear second--order partial differential equations are found. As a by-product of our results, new families of orthogonal polynomials are derived. Our approach is also applicable to dynamical systems that are not multi--particle by their nature but that can be regarded as multi--particle (for example, the Darboux--Halphen system and its generalizations). A wide class of two and three--particle polynomial dynamical systems is integrated

Power and non-power expansions of the solutions for the fourth-order analogue to the second Painlev\'{e} equation

Fourth - order analogue to the second Painlev\'{e} equation is studied. This equation has its origin in the modified Korteveg - de Vries equation of the fifth order when we look for its self - similar solution. All power and non - power expansions of the solutions for the fouth - order analogue to the second Painlev\'{e} equation near points $z=0$ and $z=\infty$ are found by means of the power geometry method. The exponential additions to solutions of the equation studied are determined. Comparison of the expansions found with those of the six Painlev\'{e} equations confirm the conjecture that the fourth - order analogue to the second Painlev\'{e} equation defines new transcendental functions.Comment: 34 pages, 8 figures; submitted to Chaos,Solitons & Fractal

Non-equilibrium critical relaxation of the 3D Heisenberg magnets with long-range correlated disorder

Monte Carlo simulations of the short-time dynamic behavior are reported for three-dimensional Heisenberg model with long-range correlated disorder at criticality, in the case corresponding to linear defects. The static and dynamic critical exponents are determined for systems starting from an ordered initial state. The obtained values of the exponents are in a good agreement with results of the field-theoretic description of the critical behavior of this model in the two-loop approximation.Comment: 14 PTPTeX pages, 10 figures. arXiv admin note: substantial text overlap with arXiv:0709.0997, arXiv:1005.521

Large and symmetric: The Khukhro--Makarenko theorem on laws --- without laws

We prove a generalisation of the Khukhro--Makarenko theorem on large characteristic subgroups with laws. This general fact implies new results on groups, algebras, and even graphs and other structures. Concerning groups, we obtain, e.g., a fact in a sense dual to the Khukhro--Makarenko theorem. A graph-theoretic corollary is an analogue of this theorem in which planarity plays the role of a multilinear identity. We answer also a question of Makarenko and Shumyatsky.Comment: 12 pages, 3 figures. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V4: misprints correcte

Polygons for finding exact solutions of nonlinear differential equations

New method for finding exact solutions of nonlinear differential equations is presented. It is based on constructing the polygon corresponding to the equation studied. The algorithms of power geometry are used. The method is applied for finding one -- parameter exact solutions of the generalized Korteveg -- de Vries -- Burgers equation, the generalized Kuramoto - Sivashinsky equation, and the fifth -- order nonlinear evolution equation. All these nonlinear equations contain the term $u^mu_x$. New exact solitary waves are found.Comment: 16 pages, 2 figure

Dimensional effects in ultrathin magnetic films

Dimensional effects in the critical properties of multilayer Heisenberg films have been numerically studied by Monte Carlo methods. The effect of anisotropy created by the crystal field of a substrate has been taken into account for films with various thicknesses. The calculated critical exponents demonstrate a dimensional transition from two-dimensional to three-dimensional properties of the films with an increase in the number of layers. A spin-orientation transition to a planar phase has been revealed in films with thicknesses corresponding to the crossover region.Comment: 5 LaTeX pages, 6 figure

On resolvability of Lindel\"of generated spaces

In this paper we study the properties of P-generated spaces (by analogy with compactly generated). We prove that a regular Lindel\"of generated space with uncountable dispersion character is resolvable. It is proved that Hausdorff hereditarily L-spaces are L-tight spaces which were defined by Istv\'an Juh\'asz, Jan van Mill in (Variations on countable tightness, arXiv:1702.03714v1). We also prove {\omega}-resolvability of regular L-tight space with uncountable dispersion character.Comment: 11 pages, 1 figur

Cosmology with nonminimal kinetic coupling and a power-law potential

We consider cosmological dynamics in the theory of gravity with the scalar field possessing a nonminimal kinetic coupling to gravity, $\kappa G_{\mu\nu}\phi^{\mu}\phi^{\nu}$, and the power-law potential $V(\phi)=V_0\phi^N$. Using the dynamical system method, we analyze all possible asymptotical regimes of the model under investigation and show that for sloping potentials with $0<N<2$ there exists a quasi-de Sitter asymptotic $H={1}/{\sqrt{9\kappa}}$ corresponding to an early inflationary Universe. In contrast to the standard inflationary scenario, the kinetic coupling inflation does not depend on a scalar field potential and is only determined by the coupling parameter $\kappa$. We obtain that there exist two different late-time asymptotical regimes. The first one leads to the usual power-like cosmological evolution with $H=1/3t$, while the second one represents the late-time inflationary Universe with $H=1/\sqrt{3\kappa}$. This secondary inflationary phase depends only on $\kappa$ and is a specific feature of the model with nonminimal kinetic coupling. Additionally, an asymptotical analysis shows that for the quadric potential with N=2 the asymptotical regimes remain qualitatively the same, while the kinetic coupling inflation is impossible for steep potentials with N>2. Using a numerical analysis, we also construct exact cosmological solutions and find initial conditions leading to the initial kinetic coupling inflation followed either by a "graceful" oscillatory exit or by the secondary inflation.Comment: 10 pages, 6 figures, submitted to PR