Special polynomials associated with rational solutions of some hierarchies

New special polynomials associated with rational solutions of the Painlevé hierarchies are introduced. The Hirota relations for these special polynomials are found. Differential-difference hierarchies to find special polynomials are presented. These formulae allow us to search special polynomials associated with the hierarchies. It is shown that rational solutions of the Caudrey–Dodd–Gibbon, the Kaup–Kupershmidt and the modified hierarchy for these ones can be obtained using new special polynomials.     

Rational solutions of the classical Boussinesq system

Rational solutions of the classical Boussinesq system are expressed in terms of special polynomials associated with rational solutions of the fourth Painlevé equation, which arises as a scaling reduction of the classical Boussinesq system. Generalized rational solutions of the classical Boussinesq system, which involve an infinite number of arbitrary constants, are also derived. The generalized rational solutions are analogues of such solutions for the Korteweg–de Vries, Boussinesq and nonlinear Schrödinger equations.     

Justification of Asymptotic Formulas for the Fourth Painlevé Equation

A special case of the fourth Painlevé equation is studied. The existence theorem is proved, and asymptotic formulas for the two parametric family of solutions near negative infinity are obtained.     

Bäcklund Transformations and Solution Hierarchies for the Third Painlevé Equation

In this article our concern is with the third Painlevé equation
d² y /d x ²= (1/ y )(d y /d x )²− (1/ x )(d y /d x ) + ( αy ²+ β )/ x + γy ³+ δ / y
where α, β, γ, and δ are arbitrary constants. It is well known that this equation admits a variety of types of solution and here we classify and characterize many of these. Depending on the values of the parameters the third Painlevé equation can admit solutions that may be either expressed as the ratio of two polynomials in either x or x ^1/3 or related to certain Bessel functions. It is thought that all exact solutions of (1) can be categorized into one or other of these hierarchies. We show how, given a few initial solutions, it is possible to use the underlying structures of these hierarchies to obtain many other solutions. In addition, we show how this knowledge concerning the continuous third Painlevé equation (1) can be adapted and used to derive exact solutions of a suitable discretized counterpart of (1). Both the continuous and discrete solutions we find are of potential importance as it is known that the third Painlevé equation has a large number of physically significant applications.     

Painlevé IV Asymptotics for Orthogonal Polynomials with Respect to a Modified Laguerre Weight

We study polynomials that are orthogonal with respect to the modified Laguerre weight   z ^{− n +ν} e ^{− Nz} ( z − 1)^{2 b}   , in the limit where   n , N →∞  with   N / n → 1  and ν is a fixed number in     . With the effect of the factor (   z − 1)^{2 b}   , the local parametrix near the critical point z = 1 can be constructed in terms of Ψ functions associated with the Painlevé IV equation. We show that the asymptotics of the recurrence coefficients of orthogonal polynomials can be described in terms of specified solution of the Painlevé IV equation in the double scaling limit. Our method is based on the Deift/Zhou steepest decent analysis of the Riemann–Hilbert problem associated with orthogonal polynomials.     

Chazy Classes IX–XI Of Third-Order Differential Equations

In this article, we study Classes IX–XI of the 13 classes introduced by Chazy (1911) in his classification of third-order differential equations in the polynomial class having the Painlevé property. Classes IX and X are the only Chazy classes that have remained unsolved to this day, and they have been at the top of our "most wanted" list for some time. (There is an incorrect claim in the literature that these classes are unstable.) Here we construct their solutions in terms of hyperelliptic functions of genus 2, which are globally meromorphic. (We also add a parameter to Chazy Class X, overlooked in Chazy's original paper.) The method involves transforming to a more tractable class of fourth- and fifth-order differential equations, which is the subject of an accompanying paper (paper I). Most of the latter equations involve hyperelliptic functions and/or higher-order Painlevé transcendents. In the case of Chazy Class XI, the solution is elementary and well known, but there are interesting open problems associated with its coefficient functions, including the appearance of one of the aforementioned transcendents. In an appendix, we present the full list of Chazy equations (in the third-order polynomial class) and the solutions of those that are not dealt with in the body of this article.     

Soliton,rational and special solutions of the Korteweg–de Vries hierarchy

Soliton, rational and special solutions for any member of the Korteweg–de Vries hierarchy are presented. It is shown that the soliton solutions are found by means of the Hirota formulae with change of the dispersion relation, special solutions for the Korteweg–de Vries hierarchy are expressed via the transcendents of the first and the second Painlevé hierarchies and rational solutions of members for the Korteveg–de Vries hierarchy can be presented using the Yablonski–Vorobév polynomials.     

Point vortices and polynomials of the Sawada-Kotera and Kaup-Kupershmidt equations

Rational solutions and special polynomials associated with the generalized K ₂ hierarchy are studied. This hierarchy is related to the Sawada-Kotera and Kaup-Kupershmidt equations and some other integrable partial differential equations including the Fordy-Gibbons equation. Differential-difference relations and differential equations satisfied by the polynomials are derived. The relationship between these special polynomials and stationary configurations of point vortices with circulations Γ and −2Γ is established. Properties of the polynomials are studied. Differential-difference relations enabling one to construct these polynomials explicitly are derived. Algebraic relations satisfied by the roots of the polynomials are found.     

Higher-order Painlevé Equations in the Polynomial Class I. Bureau Symbol P2

In this article, we construct all fourth- and fifth-order differential equations in the polynomial class having the Painlevé property and having the Bureau symbol P 2. The fourth-order equations (including the Bureau barrier equation, y ^(iv)=3 yy "−4( y ')², which fails some Painlevé tests) are six in number and are denoted F-I,…,F-VI; the fifth-order equations are four in number and are denoted Fif-I,…,Fif-IV. The 12 remaining equations of the fourth order in the polynomial class (where the Bureau symbol is P 1) are listed in the Appendix, their proof of uniqueness being postponed to a sequel (paper II). Earlier work on this problem by Bureau, Exton, and Martynov is incomplete, Martynov having found 13 of the 17 distinct reduced equations. Equations F-VI and Fif-IV are new equations defining new higher-order Painlevé transcendents. Other higher-order transcendents appearing here may be obtained by group-invariant reduction of the KdV5, Sawada–Kotera, and Kaup–Kupershmidt equations, the latter two being related. Four sections are devoted to solutions, first integrals, and assorted properties of the main equations. Several of the equations are solved in terms of hyperelliptic functions of genus 2 by means of Jacobi's postmultiplier theory. Except for a classic solution of Drach, we believe that all of these hyperelliptic solutions are new. In an accompanying paper, the hyperelliptic solutions of F-V and F-VI are applied to the unsolved third-order Chazy classes IX and X.     

ON THE COEFFICIENTS OF DIFFERENTIATED EXPANSIONS AND DERIVATIVES OF CHEBYSHEV POLYNOMIALS OF THE THIRD AND FOURTH KINDS

Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds themselves are proved.Two other explicit formulae which express the third and fourth kinds Chebyshev expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of their original expansion coefficients are also given.Two new reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced.As an application of how to use Chebyshev polynomials of the third and fourth kinds for solving high-order boundary value problems,two spectral Galerkin numerical solutions of a special linear twelfth-order boundary value problem are given.     

Rational ODEs with Movable Algebraic Singularities

A class of second-order rational ordinary differential equations, admitting certain families of formal algebraic series solutions, is considered. For all solutions of these equations, it is shown that any movable singularity that can be reached by analytic continuation along a finite-length curve is an algebraic branch point. The existence of these formal series expansions is straightforward to determine for any given equation in the class considered. We apply the theorem to a family of equations, admitting different kinds of algebraic singularities. As a further application we recover the known fact for generic values of parameters that the only movable singularities of solutions of the Painlevé equations   P_II – P   _VI   are poles.     

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.     

Cyclic Maya diagrams and rational solutions of higher order Painlevé systems

This paper focuses on the construction of rational solutions for the -Painlevé system, also called the Noumi-Yamada system, which are considered the higher order generalizations of P_IV. In this even case, we introduce a method to construct the rational solutions based on cyclic dressing chains of Schrödinger operators with potentials in the class of rational extensions of the harmonic oscillator. Each potential in the chain can be indexed by a single Maya diagram and expressed in terms of a Wronskian determinant whose entries are Hermite polynomials. We introduce the notion of cyclic Maya diagrams and characterize them for any possible period, using the concepts of genus and interlacing. The resulting classes of solutions can be expressed in terms of special polynomials that generalize the families of generalized Hermite, generalized Okamoto, and Umemura polynomials, showing that they are particular cases of a larger family.     

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.     

Nonlocal Symmetries and Explicit Solutions of the Boussinesq Equation

The nonlocal symmetry of the Boussinesq equation is obtained from the known Lax pair. The explicit analytic interaction solutions between solitary waves and cnoidal waves are obtained through the localization procedure of nonlocal symmetry. Some other types of solutions, such as rational solutions and error function solutions, are given by using the fourth Painlev′e equation with special values of the parameters. For some interesting solutions, the figures are given out to show their properties.     

Higher-Order Painlevé Equations in the Polynomial Class II: Bureau Symbol P1

In this article, we complete the Painlevé classification of fourth-order differential equations in the polynomial class that was begun in paper I, where the subcase having Bureau symbol P 2 was treated. This article treats the more difficult subcase having Bureau symbol P 1. Some of the calculations involve the use of computer searches to find all cases of integer resonances. Other cases are better handled with the Conte–Fordy–Pickering test for negative resonances. The final list consists of 19 equations denoted F-I, F-II, … , F-XIX, 17 of which have the Painlevé property while 2 (F-II, F-XIX) have Painlevé violations but are nevertheless very interesting from the point of view of Painlevé analysis. The main task of this article is to prove that the 17 Painlevé-type equations and the equivalence classes that they generate provide the complete classification of the fourth-order polynomial class. Equations F-V, F-VI, F-XVII, and F-XVIII define higher-order Painlevé transcendents. Of these, F-VI was new in paper I while the other three are group-invariant reductions of the KdV5, the modified KdV5, and the modified Sawada–Kotera equations, respectively. Seven of the 19 equations involve hyperelliptic functions of genus 2. Partial results on the fourth-order classification problem have been obtained previously by Bureau, Exton, and Martynov, the latter author finding all but four of the relevant reduced equations. Complete solutions are given except in the cases that define the aforementioned higher-order transcendents.     

Binomial-Type Ordinary Differential Equations of the Third Order

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

Generalized Adler-Moser and Loutsenko polynomials for point vortex equilibria

Equilibrium configurations of point vortices with circulations of two discrete values are associated with the zeros of a sequence of polynomials having many continuous parameters: the Adler-Moser polynomials in the case of circulation ratio ?1, and the Loutsenko polynomials in the case of ratio ?2. In this paper a new set of polynomial sequences generalizing the vortex system to three values of circulations is constructed. These polynomials are extensions of the previously known polynomials in the sense that they are special cases of the latter when the third circulation is zero. The polynomials are naturally connected with rational functions that satisfy a second-order differential equation.