Uniqueness theorems for solutions of Painlevé transcendents

The paper deals with the uniqueness problems when two meromorphic functions f and g share three distinct values CM and f satisfies the first, second or fourth Painlevé transcendents.     

A method for calculating the Painlevé transcendents

A numerical method for solving the Cauchy problem for all the six Painlevé equations is proposed. The difficulty of solving these equations is that the unknown functions can have movable (that is, dependent on the initial data) singular points of the pole type. Moreover, the Painlevé III–VI equations may have singularities at points where the solution takes certain finite values. The positions of all 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. Such auxiliary equations are derived for all Painlevé equations and for all types of singularities. Efficient criteria for transition to auxiliary systems are formulated, and numerical results illustrating the potentials of the method are presented.     

Numerical solution of the Cauchy problem for the Painlevé I and II equations

A numerical method for solving the Cauchy problem for the first and second Painlevé differential equations is proposed. The presence of movable poles of the solution is allowed. The positions of the poles are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to an auxiliary system of differential equations in a neighborhood of a pole. The equations in this system and its solution have no singularities in either the pole or its neighborhood. Numerical results confirming the efficiency of this method are presented.     

Padé approximations to certain power series

Summary An explicit identity involvingQ _n (q ⁱ z) (i = 0, 1,, 4) is shown, whereQ _n (z) is the denominator of thenth Padé approximant to the functionf(z) = _k=0 q ^1/2k(k–1 Z ^k . By using the Padé approximations, irrationality measures for certain values off(z) are also given.

     

Matrix Painlevé II equations

Theoretical and Mathematical Physics - We use the Painlevé–Kovalevskaya test to find three matrix versions of the Painlevé II equation. We interpret all these equations as...     

Generalized Padé approximations to the exponential function

The stability properties of the Padé rational approximations to the exponential function are of importance in determining the linear stability properties of several classes of Runge-Kutta methods. It is well known that the Padé approximationR _n,m (z) =N _n,m (z)/M _n,m (z), whereN _n,m (z) is of degreen andM _n,m (z) is of degreem, is A-stable if and only if 0 m – n 2, a result first conjectured by Ehle. In the study of the linear stability properties of the broader class of general linear methods one must generalize these rational approximations. In this paper we introduce a generalization of the Padé approximations to the exponential function and present a method of constructing these approximations for arbitrary order and degree. A generalization of the Ehle inequality is considered and, in the case of the quadratic Padé approximations, evidence is presented that suggests the inequality is both necessary and sufficient for A-stability. However, in the case of the cubic Padé approximations, the inequality is shown to be insufficient for A-stability. A generalization of the restricted Padé approximation, in which the denominator has a singlem-fold zero, is also introduced. A procedure for the construction of these restricted approximations is described, and results are presented on the A-stability of the restricted quadratic Padé approximations. Finally, to demonstrate the connection between a generalized Padé approximation and a general linear method, a specific general linear method is constructed with a stability region corresponding to a given quadratic Padé approximation.     

Some properties of biorthogonal polynomials and their application to Padé approximations

Transformations of biorthogonal polynomials under certain transformations of biorthogonalizable sequences are studied. The obtained result is used to construct Padé approximants of orders [N?1/N],N ε ?, for the functions $$\tilde f(z) = \sum\limits_{m = 0}^M {\alpha _m } \frac{{f(z) - T_{m - 1} [f;z]}}{{z^m }},$$ wheref(z) is a function with known Padé approximants of the indicated orders,T _j [f;z] are Taylor polynomials of degreej for the functionf(z), and α _{m, M} = $\overline {1,M} $ are constants.     

“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.     

Distributions of Poles to Painlevé Transcendents via Padé Approximations

A version of the Fair–Luke algorithm has been used to find the Padé approximate solutions to the Painlevé I, II, and IV equations. The distributions of poles in the complex plane are studied to check the dynamics of movable poles and the emergence of rational and truncated solutions, as well as various patterns formed by the poles. The high-order approximations allow us to check asymptotic expansions at infinity and estimate the range of asymptotic domains. The Coulomb gas interpretation of the pole ensembles is discussed in view of the patterns arising in Painlevé IV transcendents.     

Higher Painlevé transcendents as special solutions of some nonlinear integrable hierarchies

It is well known that the self-similar solutions of the Korteweg-de Vries equation and the modified Korteweg-de Vries equation are expressed via the solutions of the first and second Painlevé equations. In this paper we solve this problem for all equations from the Korteveg-de Vries, modified Korteweg-de Vries, Kaup-Kupershmidt, Caudrey-Dodd-Gibbon and Fordy-Gibbons hierarchies. We show that the self-similar solutions of equations corresponding to hierarchies mentioned above can be found by means of the general solutions of higher-order Painlevé hierarchies introduced more than ten years ago.     

Discrete Painlevé I and singularity confinement in projective space

We write the discrete Painlevé I equation as a polynomial map in projective space and follow the development of the singularity and the fate of the other initial value. Most of the time the initial value is at the next to leading term and we can recover it (and confine the singularity) at the (3m+1)th iteration by imposing a condition on the de-autonomization. For m=1 one gets the usual d-P_I.     

Convergence of subdiagonal Padé approximations of C 0-semigroups

Let ${(r_{n})_{n \in \mathbb{N}}}$ be the sequence of subdiagonal Padé approximations of the exponential function. We prove that for ?A the generator of a uniformly bounded C ₀-semigroup T on a Banach space X, the sequence ${(r_{n}(-t A))_{n \in \mathbb{N}}}$ converges strongly to T(t) on D(A ^α ) for ${\alpha>\frac{1}{2}}$ . Local uniform convergence in t and explicit convergence rates in n are established. For specific classes of semigroups, such as bounded analytic or exponentially γ -stable ones, stronger estimates are proved. Finally, applications to the inversion of the vector-valued Laplace transform are given.     

On Painlevé's equations I, II and IV

We give a new proof of the fact that the solutions of Painlevé's differential equations I, II and IV are meromorphic functions in the complex plane. The method of proof is based on differential inequality techniques.     

Padé and Padé-Type Approximation for 2π-Periodic Lp Functions

The numerical evaluation of a 2-periodic L ^p function by its Fourier series expansion may become a difficult task whenever only a few coefficients of this series are known or it converges too slowly. In this paper we propose a general method to evaluate such any function, by means of composed Padé-type approximants. The definition, the main ideas, and the properties of the approximants will be given. After having done this successfully, we will consider several concrete examples and a theoretical application to the convergence acceleration problem of functional sequences.     

Monodromy of certain Painlevé–VI transcendents and reflection groups

We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters β=γ=0, δ= and 2α=(2μ-1)² with arbitrary μ, 2μ≠∈ℤ. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. We show that the finite orbits of this action correspond to the algebraic solutions of our Painlevé VI equation and use this result to classify all of them. We prove that the algebraic solutions of our Painlevé VI equation are in one-to-one correspondence with the regular polyhedra or star-polyhedra in the three dimensional space. Oblatum 19-III-1999 & 25-XI-1999?Published online: 21 February 2000     

The convergence of diagonal Padé approximants and the Padé Conjecture

Questions related to the convergence problem of diagonal Padé approximants are discussed. A central place is taken by the Padé Conjecture (also known as the Baker-Gammel-Wills Conjecture). Partial results concerning this conjecture are reviewed and weaker and more special versions of the conjecture are formulated and their plausibility is investigated. Great emphasis is given to the role of spurious poles of the approximants. A conjecture by Nuttall (1970) about the number and distribution of such poles is stated and its importance for the Padé Conjecture is analyzed.