Hamiltonians associated with the sixth Painlevé equation

We obtain the formula determining the general form of polynomial Hamiltonians associated with the sixth Painlevé equation and prove its uniqueness. We prove the existence of nonpolynomial Hamiltonians associated with this equation. We identify the Hamiltonian class for which the defining differential equation coincides with the equation (h-equation) for the simplest polynomial Hamiltonian (the Okamoto Hamiltonian). __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 54–65, April, 2007.     

The Painlevé III equation and the Iwasawa decomposition

For the third Painlevé equation an explicit isomorphism between the monodromy data and the data of the approach of Dorfmeister-Pedit-Wu, based on the Iwasawa decomposition of the loop groups, is established. As an application, this provides a simple algebraic way to calculate the monodromy data in terms of the Cauchy data at zero.

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Partially supported by the Sonderforschungsbereich 288

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Partially supported by the National Science Foundation (DMS-9315964)

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This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag.     

Asymptotic description of solutions of the fourth Painlevé equations on the stokes rays analogs

The proof of the existence of classical solutions is given for the Painlevé type nonlinear ordinary differential equations. The solutions have asymptotic formulas, which can be obtained by the isomonodromic deformation method.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 101–109, 1989.     

Expansion of solutions of the sixth Painlevé equation near a regular point

For the sixth Painlevé equation, by using power-geometry methods, all asymptotic expansions of solutions are obtained in a neighborhood of a regular point of the independent variable. All of these expansions are convergent series in integer powers with constant complex coefficients. Five families of expansions are found. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 36, Suzdal Conference-2004, Part 2, 2005.     

Special polynomials and rational solutions of the hierarchy of the second Painlevé equation

We study special polynomials used to represent rational solutions of the hierarchy of the second Painlevé equation. We find several recursion relations satisfied by these polynomials. In particular, we obtain a differential-difference relation that allows finding any polynomial recursively. This relation is an analogue of the Toda chain equations. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 1, pp. 58–67, October, 2007.     

Painlevé Hierarchies and the Painlevé Test

In a series of recent papers, we derived several new hierarchies of higher-order analogues of the six Painlevé equations. Here we consider one particular example of such a hierarchy, namely, a recently derived fourth Painlevé hierarchy. We use this hierarchy to illustrate how knowing the Hamiltonian structures and Miura maps can allow finding first integrals of the ordinary differential equations derived. We also consider the implications of the second member of this hierarchy for the Painlevé test. In particular, we find that the Ablowitz–Ramani–Segur algorithm cannot be applied to this equation. This represents a significant failing in what is now a standard test of singularity structure. We present a solution of this problem.     

Integral transformation of solutions for a Fuchsian-class equation corresponding to the Okamoto transformation of the Painlevé VI equation

We show that under the Euler integral transformation with the kernel (x−z)^−α, some solutions of the Fuchs equations (the original pair for the Painlevé VI equation) pass into solutions of a system of the same form with the parameters changed according to the Okamoto transformation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 355–364, March, 2006.     

On the convergence of a fourth order evolution equation to the Allen-Cahn equation

We construct special sequences of solutions to a fourth order nonlinear parabolic equation of Cahn-Hilliard/Allen-Cahn type, converging to the second order Allen-Cahn equation. We consider the evolution equation without boundary, as well as the stationary case on domains with Dirichlet boundary conditions. The proofs exploit the equivalence of the fourth order equation with a system of two second order elliptic equations with “good signs”.     

The second Painlevé equation in a problem concerning nonlinear effects near caustics

In a critical case we study the asymptotics (at large time) of a solution of the nonlinear Schrödinger equation. This solution arises in a series of problems when accounting for the nonlinear effects near caustics. The asymptotics is described in terms of the second Painlevé transcendent. Bibliography: 34 titles.     

Expansion of solutions to the fifth Painlevé equation in a neighborhood of a nonsingular point

For the fifth Painlevé equation, the asymptotic expansions of solutions are obtained in a neighborhood of a nonsingular point of the independent variable by using methods of power geometry. Four families of expansions are found. All of these expansions are convergent series in integral powers with complex coefficients. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 36, Suzdal Conference-2004, Part 2, 2005.     

Numerical integration of the landau-lifshits-gilbert equation

An efficient projection-difference method is proposed for numerical integration of the equation of motion of magnetization, based on the weak form of the equation.Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 155–159, 1986.     

On the stability of a nonhomogeneous differential equation of the fourth order

Summary In previous papersEzeilo [3],Harrow [4, 5] had established stability results for the equations(1.3), (1.4) and(1.5). In the present paper these results are extended to hold for the equation(1.1). Entrata in Redazione il 16 novembre 1970.     

Boundedness character of a fourth order nonlinear difference equation

In this paper, among others, we give a complete picture of the boundedness character of the positive solutions of the following nonlinear difference equation

where the parameters A and p are positive real numbers. This is a natural extension of a difference equation equivalent to an equation arising in automatic control theory. We present some new methods for investigating the boundedness character of nonlinear difference equations.