Hamiltonians associated with the third and fifth Painlevé equations

We obtain a Painlevé-type differential equation for the simplest rational Hamiltonian associated with the fifth Painlevé equation in the case γ ≠ 0, δ = 0. We prove the existence of Hamiltonians of a nonrational type associated with the fifth Painlevé equation in the case γ ≠ 0, δ = 0. We obtain a generalization of the Garnier and Okamoto formulas for rational Hamiltonians associated with the third Painlevé tequation.     

Picard and Chazy solutions to the Painlevé VI equation

We study the solutions of a particular family of Painlevé VI equations with parameters and , for . We show that in the case of half-integer , all solutions can be written in terms of known functions and they are of two types: a two-parameter family of solutions found by Picard and a new one-parameter family of classical solutions which we call Chazy solutions. We give explicit formulae for them and completely determine their asymptotic behaviour near the singular points and their nonlinear monodromy. We study the structure of analytic continuation of the solutions to the PVI equation for any such that . As an application, we classify all the algebraic solutions. For half-integer, we show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For integer, we show that all algebraic solutions belong to a one-parameter family of rational solutions. Received: 23 February 1999 / Accepted: 10 January 2001 / Published online: 18 June 2001     

Numerical solution of the Painlevé VI equation

A numerical method for solving the Cauchy problem for the sixth Painlevé equation is proposed. The difficulty of this problem, as well as the other Painlevé equations, is that the unknown function can have movable singular points of the pole type; moreover, the equation may have singularities at the points where the solution takes the values 0 or 1 or is equal to the independent variable. The positions of all of these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. The main results of this paper are the derivation of the auxiliary equations and the formulation of transition criteria. Numerical results illustrating the potentials of this method are presented.     

Numerical solution of the Cauchy problem for Painlevé III

We suggest a numerical method for solving the Cauchy problem for the third Painlevé equation. The solution of this problem is complicated by the fact that the unknown function can have movable singular points of the pole type, and in addition, the equation has a singularity at the points where the solution vanishes. The position of poles and zeros of the function is not given and is specified in the course of the solution. The method is based on the passage, in a neighborhood of these points, to an auxiliary system of differential equations for which the equation and the corresponding solution has no singularity in that neighborhood and at the pole or zero itself. We present the results of numerical experiments, which justify the efficiency of the suggested method.     

Painlevé analysis and exact solutions of the Korteweg–de Vries equation with a source

We consider the Korteweg–de Vries equation with a source. The source depends on the solution as polynomials with constant coefficients. Using the Painlevé test we show that the generalized Korteweg–de Vries equation is not integrable by the inverse scattering transform. However there are some exact solutions of the generalized Korteweg–de Vries equation for two forms of the source. We present these exact solutions.     

Darboux polynomials,balances and Painlevé property

For a given polynomial differential system we provide different necessary conditions for the existence of Darboux polynomials using the balances of the system and the Painlevé property. As far as we know, these are the first results which relate the Darboux theory of integrability, first, to the Painlevé property and, second, to the Kovalevskaya exponents. The relation of these last two notions to the general integrability has been intensively studied over these last years.     

“Quantizations” of higher Hamiltonian analogues of the Painlevé I and Painlevé II equations with two degrees of freedom

We construct a solution of an analogue of the Schrödinger equation for the Hamiltonian H ₁(z, t, q ₁, q ₂, p ₁, p ₂) corresponding to the second equation P ₁ ² in the Painlevé I hierarchy. This solution is obtained by an explicit change of variables from a solution of systems of linear equations whose compatibility condition is the ordinary differential equation P ₁ ² with respect to z. This solution also satisfies an analogue of the Schrödinger equation corresponding to the Hamiltonian H ₂(z, t, q ₁, q ₂, p ₁, p ₂) of a Hamiltonian system with respect to t compatible with P ₁ ² . A similar situation occurs for the P ₂ ² equation in the Painlevé II hierarchy.     

Hardy and Cowling–Price theorems associated with the Jacobi–Cherednik operator

This paper is devoted to the proof of Hardy and Cowling–Price type theorems for the Fourier transform tied to the Jacobi–Cherednik operator.     

Gradient blowup rate for a viscous Hamilton–Jacobi equation with degenerate diffusion

This paper is concerned with the gradient blowup rate for the one-dimensional p-Laplacian parabolic equation ${u_t=(|u_x|^{p-2} u_x)_x +|u_x|^q}$ with q > p > 2, for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We establish the blowup rate estimates of lower and upper bounds and show that in this case the blowup rate does not match the self-similar one.     

Surfaces with Harmonic Inverse Mean Curvature and Painlevé Equations

In this paper we study surfaces immersed in R³ such that the mean curvature function H satisfies the equation (1/H) = 0, where is the Laplace operator of the induced metric. We call them HIMC surfaces. All HIMC surfaces of revolution are classified in terms of the third Painlevé transcendent. In the general class of HIMC surfaces we distinguish a subclass of -isothermic surfaces, which is a generalization of the isothermic HIMC surfaces, and classify all the -isothermic HIMC surfaces in terms of the solutions of the fifth and sixth Painlevé transcendents.     

Continued fractions associated with the Newton-Padé table

Summary We discuss first the block structure of the Newton-Padé table (or, rational interpolation table) corresponding to the double sequence of rational interpolants for the data{(z _k, h(z_k)} _k =0. (The (m, n)-entry of this table is the rational function of type (m,n) solving the linearized rational interpolation problem on the firstm+n+1 data.) We then construct continued fractions that are associated with either a diagonal or two adjacent diagonals of this Newton-Padé table in such a way that the convergents of the continued fractions are equal to the distinct entries on this diagonal or this pair of diagonals, respectively. The resulting continued fractions are generalizations of Thiele fractions and of Magnus'sP-fractions. A discussion of an some new results on related algorithms of Werner and Graves-Morris and Hopkins are also given.Dedicated to the memory of Helmut Werner (1931–1985)     

On the relation between the continuous and discrete Painlevé equations

A method for deriving difference equations (the discrete Painlevé equations in particular) from the Bäcklund transformations of the continuous Painlevé equations is discussed. This technique can be used to derive several of the known discrete painlevé equations (in particular, the first and second discrete Painlevé equations and some of their alternative versions). The Painlevé equations possess hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions for special values of the parameters. Hence, the aforementioned relations can be used to generate hierarchies of exact solutions for the associated discrete Painlevé equations. Exact solutions of the Painlevé equations simultaneously satisfy both a differential equation and a difference equation, analogously to the special functions.     

Potential operators associated with Jacobi and Fourier–Bessel expansions

We study potential operators (Riesz and Bessel potentials) associated with classical Jacobi and Fourier–Bessel expansions. We prove sharp estimates for the corresponding potential kernels. Then we characterize those 1≤p,q≤∞$1\le p,q\le \infty$, for which the potential operators are of strong type (p,q)$(p,q)$, of weak type (p,q)$(p,q)$ and of restricted weak type (p,q)$(p,q)$. These results may be thought of as analogues of the celebrated Hardy–Littlewood–Sobolev fractional integration theorem in the Jacobi and Fourier–Bessel settings. As an ingredient of our line of reasoning, we also obtain sharp estimates of the Poisson kernel related to Fourier–Bessel expansions.