39,165 research outputs found
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 and 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 . 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, , and the power-law potential
. Using the dynamical system method, we analyze all possible
asymptotical regimes of the model under investigation and show that for sloping
potentials with there exists a quasi-de Sitter asymptotic
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 . We obtain that there exist two different late-time
asymptotical regimes. The first one leads to the usual power-like cosmological
evolution with , while the second one represents the late-time
inflationary Universe with . This secondary inflationary
phase depends only on 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
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