Universal characters, integrable chains and the Painlevé equations

The universal character is a generalization of the Schur polynomial attached to a pair of partitions; see (Adv. Math. 74 (1989) 57). We prove that the universal character solves the Darboux chain. The N-periodic closing of the chain is equivalent to the Painlevé equation of type . Consequently we obtain an expression of rational solutions of the Painlevé equations in terms of the universal characters.     

Autonomous self-similar ordinary differential equations and the Painlevé connection

We demonstrate an intimate connection between nonlinear higher-order ordinary differential equations possessing the two symmetries of autonomy and self-similarity and the leading-order behaviour and resonances determined in the application of the Painlevé Test. Similar behaviour is seen for systems of first-order differential equations. Several examples illustrate the theory. In an integrable case of the ABC system the singularity analysis reveals a positive and a negative resonance and the method of leading-order behaviour leads naturally to a Laurent expansion containing both.     

Painlevé Classification of Binomial Type Ordinary Differential Equations of the Arbitrary Order

The complete Painlevé classification of the binomial ordinary differential equations of the arbitrary order n ≥ 4 is built. Six classes of equations with Painlevé property are obtained. All of these equations are solved in terms of elementary functions and known Painlevé transcendents.     

Special functions arising from discrete Painlevé equations: A survey

This article is a survey on recent studies on special solutions of the discrete Painlevé equations, especially on hypergeometric solutions of the q-Painlevé equations. The main part of this survey is based on the joint work [K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada, Hypergeometric solutions to the q-Painlevé equations, IMRN 2004 47 (2004) 2497–2521, K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada, Construction of hypergeometric solutions to the q-Painlevé equations, IMRN 2005 24 (2005) 1439–1463] with Kajiwara, Masuda, Ohta and Yamada. After recalling some basic facts concerning Painlevé equations for comparison, we give an overview of the present status of studies on difference (discrete) Painlevé equations as a source of special functions.     

On Discrete Painlevé Equations Associated with the Lattice KdV Systems and the Painlevé VI Equation

A new integrable nonautonomous nonlinear ordinary difference equation is presented that can be considered to be a discrete analogue of the Painlevé V equation. Its derivation is based on the similarity reduction on the two-dimensional lattice of integrable partial differential equations of Korteweg–de Vries (KdV) type. The new equation, which is referred to as generalized discrete Painlevé equation (GDP), contains various "discrete Painlevé equations" as subcases for special values/limits of the parameters, some of which have already been given in the literature. The general solution of the GDP can be expressed in terms of Painlevé VI (PVI) transcendents. In fact, continuous PVI emerges as the equation obeyed by the solutions of the discrete equation in terms of the lattice parameters rather than the lattice variables that label the lattice sites. We show that the bilinear form of PVI is embedded naturally in the lattice systems leading to the GDP. Further results include the establishment of Bäcklund and Schlesinger transformations for the GDP, the corresponding isomonodromic deformation problem, and the self-duality of its bilinear scheme.     

Value distribution of meromorphic solutions of certain difference Painlevé equations

The Borel exceptional value and the exponents of convergence of poles, zeros and fixed points of finite order transcendental meromorphic solutions for difference Painlevé I and II equations are estimated. And the forms of rational solutions of the difference Painlevé II equation and the autonomous difference Painlevé I equation are also given. It is also proved that the non-autonomous difference Painlevé I equation has no rational solution.     

Lie point symmetries, partial Noether operators and first integrals of the Painlevé-Gambier equations

We utilize the Lie-Tressé linearization method to obtain linearizing point transformations of certain autonomous nonlinear second-order ordinary differential equations contained in the Painlevé-Gambier classification. These point transformations are constructed using the Lie point symmetry generators admitted by the underlying Painlevé-Gambier equations. It is also shown that those Painlevé-Gambier equations which have a few Lie point symmetries and hence are not linearizable by this method can be integrated by a quadrature. Moreover, by making use of the partial Lagrangian approach we obtain time dependent and time independent first integrals for these Painlevé-Gambier equations which have not been reported in the earlier literature. A comparison of the results obtained in this paper is made with the ones obtained using the generalized Sundman linearization method.     

Wiener-Hope equations with the transmission property

Conditions on the kernel of the classical Wiener-Hopf equation are obtained which provide the same smoothness of any solution in terms of its inclusion in different spaces of smooth functions such as as the function in the right side of the equation.     

Integrability of a coupled KdV system: Painlevé property, Lax pair and Bäcklund transformation

The integrability of a coupled KdV system is studied by prolongation technique and singularity analysis. As a result, Bäcklund transformation and linear spectral problem associated with this system are obtained. Some special solutions of the system are also proposed.     

Paley-Wiener spaces with vanishing conditions and Painlevé VI transcendents

We modify the classical Paley-Wiener spaces PWx of entire functions of finite exponential type at most x>0, which are square integrable on the real line, via the additional condition of vanishing at finitely many complex points z₁,…,zn. We compute the reproducing kernels and relate their variations with respect to x to a Krein differential system, whose coefficient (which we call the μ-function) and solutions have determinantal expressions. Arguments specific to the case where the “trivial zeros” z₁,…,zn are in arithmetic progression on the imaginary axis allow us to establish for expressions arising in the theory a system of two non-linear first order differential equations. A computation, having this non-linear system at his start, obtains quasi-algebraic and among them rational Painlevé transcendents of the sixth kind as certain quotients of such μ-functions.     

Some examples for the Poincaré and Painlevé problems

In 1891, Poincaré started a series of three papers in which he tried to answer the following question (cf. [21-23]): “Is it possible to decide if an algebraic differential equation in two variables is algebraically integrable?” (in the sense that it has a rational first integral). More or less at the same time P. Painlevé asked the following question: “Is it possible to recognize the genus of the general solution of an algebraic differential equation in two variables which has a rational first integral?”. In this paper we give examples of one-parameter families which show that both problems have a negative answer. With some of the families we can also answer a question posed by M. Brunella in [5].     

A new derivation of Painlevé hierarchies

We give a new derivation of two Painlevé hierarchies. This is done by extending the accelerating-wave reductions of the Korteweg-de Vries and dispersive water wave equations to their respective hierarchies. We also consider the extension of this reduction of Burgers equation to the Burgers hierarchy.     

The master Painlevé VI heat equation

Given the second-order scalar Lax pair of the sixth Painlevé equation, we build a generalized heat equation with rational coefficients which does not depend any more on the Painlevé variable.     

Painlevé Classification Problems Featuring Essential Singularities

In this article we construct and solve all Painlevé-type differential equations of the second order and second degree that are built upon, in a natural well-defined sense, the "sn-log" equation of Painlevé, the general integral of which admits a movable essential singularity (elliptic function of a logarithm). This equation (which was studied by Painlevé in the years 1893–1902) is frequently cited in the modern literature to elucidate various aspects of Painlevé analysis and integrability of differential equations, especially the difficulty of detecting essential singularities by local singularity analysis of differential equations. Our definition of the Painlevé property permits movable essential singularities, provided there is no branching. While the essential singularity presents no serious technical problems, we do need to introduce new techniques for handling "exotic" Painlevé equations, which are Painlevé equations whose singular integrals admit movable branch points in the leading terms. We find that the corresponding full class of Painlevé-type equations contains three, and only three, equations, which we denote SD-326-I, SD-326-II, and SD-326-III, each solvable in terms of elliptic functions. The first is Painlevé's own generalization of his sn-log equation. The second and third are new, the third being a 15-parameter exotic master equation. The appendices contain results (in general, without uniqueness proofs) of related Painlevé classification problems, including full generalizations of two other second-degree equations discovered by Painlevé, additional examples of exotic Painlevé equations and Painlevé equations admitting movable essential singularities, and third-order equations featuring sn-log and other essential singularities.     

On tronquée solutions of the first Painlevé hierarchy

It is well known that, due to Boutroux, the first Painlevé equation admits solutions characterized by divergent asymptotic expansions near infinity in specified sectors of the complex plane. In this paper, we show that such solutions exist for higher order analogues of the first Painlevé equation (the first Painlevé hierarchy) as well.     

Bäcklund transformations for fourth Painlevé hierarchies

Bäcklund transformations (BTs) for ordinary differential equations (ODEs), and in particular for hierarchies of ODEs, are a topic of great current interest. Here, we give an improved method of constructing BTs for hierarchies of ODEs. This approach is then applied to fourth Painlevé (P_IV) hierarchies recently found by Gordoa et al. [Publ. Res. Inst. Math. Sci. (Kyoto) 37 (2001) 327-347]. We show how the known pattern of BTs for P_IV can be extended to our P_IV hierarchies. Remarkably, the BTs required to do this are precisely the Miura maps of the dispersive water wave hierarchy. We also obtain the important result that the fourth Painlevé equation has only one nontrivial fundamental BT, and not two such as is frequently stated.     

On the Lax Pairs of the Symmetric Painlevé Equations

The symmetric forms of the Painlevé equations are a sequence of nonlinear dynamical systems in N + 1 variables that admit the action of an extended affine Weyl group of type     , as shown by Noumi and Yamada. They are equivalent to the periodic dressing chains studied by Veselov and Shabat, and by Adler. In this paper, a direct derivation of the symmetries of a corresponding sequence of ( N + 1) × ( N + 1) matrix linear systems (Lax pairs) is given. The action of the generators of the extended affine Weyl group of type     on the associated Lax pairs is realized through a set of transformations of the eigenfunctions, and this extends to an action of the whole group.     

Navier–Stokes equations with regularity in one directional derivative of the pressure

This paper concerns the 3D Navier‐Stokes equations and prove an almost Serrin‐type regularity criterion in terms of one directional derivative of the pressure. Copyright © 2014 John Wiley & Sons, Ltd.