Cubic and Quartic Transformations of the Sixth Painlevé Equation in Terms of Riemann–Hilbert Correspondence

The starting point of this paper is a classification of quadratic polynomial transformations of the monodromy manifold for the 2 × 2 isomonodromic Fuchsian systems associated to the Painlevé VI equation. Up to birational automorphisms of the monodromy manifold, we find three transformations. Two of them are identified as the action of known quadratic or quartic transformations of the Painlevé VI equation. The third transformation of the monodromy manifold gives a new transformation of degree 3 of Picard’s solutions of Painlevé VI.     

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.     

Painlevé VI,Painlevé III,and the Hankel determinant associated with a degenerate Jacobi unitary ensemble

This paper studies the Hankel determinant generated by a perturbed Jacobi weight, which is closely related to the largest and smallest eigenvalue distribution of the degenerate Jacobi unitary ensemble. By using the ladder operator approach for the orthogonal polynomials, we find that the logarithmic derivative of the Hankel determinant satisfies a nonlinear second-order differential equation, which turns out to be the Jimbo–Miwa–Okamoto σ-form of the Painlevé VI equation by a translation transformation. We also show that, after a suitable double scaling, the differential equation is reduced to the Jimbo–Miwa–Okamoto σ-form of the Painlevé III. In the end, we obtain the asymptotic behavior of the Hankel determinant as t→1⁻ and t→0⁺ in two important cases, respectively.     

Quadratic transformations for the third and fifth Painlevé equations

Quadratic transformations for the third and fifth Painlev’e equations are constructed via the method of RS-transformations. This method can be viewed as a prolongation of quadratic transformations for the Painlevé equations to the associated linear ODEs, whose isomonodromy deformations are governed by the corresponding Painlevé equations. Bibliography: 15 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 317, 2004, pp. 105–121.     

λ‐symmetries,nonlocal transformations and first integrals to a class of Painlevé–Gambier equations

In this work, we consider a class of Painlevé–Gambier equations that model the motion of chain ball drawing with constant force in the frictionless surface. λ‐symmetries, first integrals, integrating factors, nonlocal transformations and local transformations are derived by using the some recent studies that are proposed by Muriel and Romero. Copyright © 2012 John Wiley & Sons, Ltd.     

Bäcklund transformations for discrete Painlevé equations: Discrete PII–PV

Transformation properties of discrete Painlevé equations are investigated by using an algorithmic method. This method yields explicit transformations which relates the solutions of discrete Painlevé equations, discrete P_II–P_V, with different values of parameters. The particular solutions which are expressible in terms of the discrete analogue of the classical special functions of discrete Painlevé equations can also be obtained from these transformations.     

Integrability and equivalence relationships of six integrable coupled Korteweg–de Vries equations

In this paper, we investigate the integrability and equivalence relationships of six coupled Korteweg–de Vries equations. It is shown that the six coupled Korteweg–de Vries equations are identical under certain invertible transformations. We reconsider the matrix representations of the prolongation algebra for the Painlevé integrable coupled Korteweg–de Vries equation in [Appl. Math. Lett. 23 (2010) 665‐669] and propose a new Lax pair of this equation that can be used to construct exact solutions with vanishing boundary conditions. It is also pointed out that all the six coupled Korteweg–de Vries equations have fourth‐order Lax pairs instead of the fifth‐order ones. Moreover, the Painlevé integrability of the six coupled Korteweg–de Vries equations are examined. It is proved that the six coupled Korteweg–de Vries equations are all Painlevé integrable and have the same resonant points, which further determines the equivalence among them. Finally, the auto‐Bäcklund transformation and exact solutions of one of the six coupled Korteweg–de Vries equations are proposed explicitly. Copyright © 2016 John Wiley & Sons, Ltd.     

Painlevé transcendents and the Hankel determinants generated by a discontinuous Gaussian weight

This paper studies the Hankel determinants generated by a discontinuous Gaussian weight with one and two jumps. It is an extension in a previous study, in which they studied the discontinuous Gaussian weight with a single jump. By using the ladder operator approach, we obtain a series of difference and differential equations to describe the Hankel determinant for the single jump case. These equations include the Chazy II equation, continuous and discrete Painlevé IV. In addition, we consider the large n behavior of the corresponding orthogonal polynomials and prove that they satisfy the biconfluent Heun equation. We also consider the jump at the edge under a double scaling, from which a Painlevé XXXIV appeared. Furthermore, we study the Gaussian weight with two jumps and show that a quantity related to the Hankel determinant satisfies a two variables' generalization of the Jimbo‐Miwa‐Okamoto σ‐form of the Painlevé IV.     

Exact Solutions of Fisher and Generalized Fisher Equations with Variable Coefficients

In this work,we consider a Fisher and generalized Fisher equations with variable coefficients.Usingtruncated Painlevé expansions of these equations,we obtain exact solutions of these equations with a constrainton the coefficients a(t)and b(t).     

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.     

Families of Okamoto–Painlevé pairs and Painlevé equations

In [as reported by Saito et al. (J. Algebraic Geom. 11:311–362, 2002)], generalized Okamoto–Painlevé pairs are introduced as a generalization of Okamoto’s space of initial conditions of Painlevé equations (cf. [Okamoto (Jpn. J. Math. 5:1–79, 1979)]) and we established a way to derive differential equations from generalized rational Okamoto–Painlevé pairs through deformation theory of nonsingular pairs. In this article, we apply the method to concrete families of generalized rational Okamoto–Painlevé pairs with given affine coordinate systems and for all eight types of such Okamoto–Painlvé pairs we write down Painlevé equations in the coordinate systems explicitly. Moreover, except for a few cases, Hamitonians associated to these Painlevé equations are also given in all coordinate charts. Mathematics Subject Classification (2000) 34M55, 32G05, 14J26     

The Generating Function for the Airy Point Process and a System of Coupled Painlevé II Equations

For a wide class of Hermitian random matrices, the limit distribution of the eigenvalues close to the largest one is governed by the Airy point process. In such ensembles, the limit distribution of the k th largest eigenvalue is given in terms of the Airy kernel Fredholm determinant or in terms of Tracy–Widom formulas involving solutions of the Painlevé II equation. Limit distributions for quantities involving two or more near‐extreme eigenvalues, such as the gap between the k th and the ℓth largest eigenvalue or the sum of the k largest eigenvalues, can be expressed in terms of Fredholm determinants of an Airy kernel with several discontinuities. We establish simple Tracy–Widom type expressions for these Fredholm determinants, which involve solutions to systems of coupled Painlevé II equations, and we investigate the asymptotic behavior of these solutions.     

Painlevé Functions and Conformal Blocks

We outline recent developments relating Painlevé equations and 2D conformal field theory. Generic tau functions of Painlevé VI and Painlevé III₃ are written as linear combinations of c=1 conformal blocks and their irregular limits. This provides explicit combinatorial series representations of the tau functions, and helps to establish a connection formula for the tau function in the Painlevé VI case.     

The Painlevé Property and Hirota's Method

The connection between the Painlevé property for partial differential equations, proposed by Weiss, Tabor, and Carnevale, and Hirota's method for calculating N-soliton solutions is investigated for a variety of equations including the nonlinear Schrödinger and mKdV equations. Those equations which do not possess the Painlevé property are easily seen not to have self-truncating Hirota expansions. The Bäcklund transformations derived from the Painlevé analysis and those determined by Hirota's method are shown to be directly related. This provides a simple route for demonstrating the connection between the singular manifolds used in the Painlevé analysis and the eigenfunctions of the AKNS inverse scattering transform.     

Painlevé Classification of All Semilinear Partial Differential Equations of the Second Order. II. Parabolic and Higher Dimensional Equations

This paper extends the work of the previous paper (I) on the Painlevé classification of second-order semilinear partial differential equations to the case of parabolic equations in two independent variables, u_xx = F(x, y, u, u_x, u_y), and irreducible equations in three or more independent variables of the form, Σ_ijR_ij (x₁,…, x_n)u,_ij = F(x₁,…, x_n; u,₁,…, u,_n). In each case, F is assumed to be rational in u and its first derivatives and no other simplifying assumptions are made. In addition to the 22 hyperbolic equations found in paper I, we find 10 equivalence classes of parabolic equations with the Painlevé property, denoted PS-I, PS-I1,…, PS-X, equation PS-II being a generalization of Burgers' equation denoted the Forsyth-Burgers equation, and 13 higher-dimensional Painlevé equations, denoted GS-I, GS-II,…, GS-XIII. The lists are complete up to the equivalence relation of Möbius transformations in u and arbitrary changes of the independent variables. In order to avoid repetition, the proofs are sketched very briefly in cases where they closely resemble those for the corresponding hyperbolic problem. Every equation is solved by transforming to a linear partial differential equation, from which it follows that there are no non trivial soliton equations among the two classes of Painlevé equations treated in this paper.     

Les Pré-(a b)-algèbres à Homotopie Près

Dans cet article on étudie le concept d'algèbre à homotopie près pour une structure définie par deux opérations ? et ?. Des exemples importants d'une telle structure sont ceux des algèbres pré-Gerstenhaber et pré-Poisson graduées. Etant donnée une structure d'algèbre pré-commutative et pré-Lie graduée pour deux décalages des degrés donnés par a et b, on définit la structure d'une pré-(a, b)-algèbre graduée et on donne une construction explicite de l'algèbre à homotopie près associée.

We study in this article the concept of algebra up to homotopy for a structure defined by two operations, ? and ?. Important examples of such structure are those of graded pre-Gerstenhaber and pre-Poisson algebras.

Given a structure of pre-commutative and pre-Lie algebra for two shifts of degree given by a and b, we define the structure of a graded pre-(a, b)-algebra, and we give an explicit construction of the associated algebra up to homotopy     

Sur la transformation des intégrales des équations différentielles linéaires ordinaires du second ordre

Résumé Le présent Mémoire contient la partie analytique d'une théorie des transformations des intégrales des équations différentielles linéaires ordinaires du second ordre. Il s'agit des équations réelles (a), (A) (p. 327) et des questions de caractère global. La théorie développée gravite autour des propriétés des équations différentielles non-linéaires du troisième ordre (b), (B) (p. 327). Sont donnés, en particulier, les théorèmes sur l'existence et l'unicité ainsi que les expressions explicites pourles intégrales des équations (b), (B).     

Rational Solutions of Painlevé Equations

With Bäcklund transformations, we construct explicit solutions of Painlevé equations 2 and 4. Independently, we find solutions of degenerate cases of equations 3 and 5. The six Painlevé transcendents are referred to as 1–6.     

All Binomial-Type Painlevé Equations of the Second Order and Degree Three or Higher

In this paper we construct all rational Painlevé-type differential equations which take the binomial form, (d²y/dx²)ⁿ = F(x,y,dy/dx), where n ≥ 3, the case n = 2 having previously been treated in Cosgrove and Scoufis [1]. While F is assumed to be rational in the complex variables y and y′ and locally analytic in x, it is shown that the Painlevé property together with the absence of intermediate powers of y″ forces F to be a polynomial in y and y′. In addition to the six classes of second-degree equations found in the aforementioned paper, we find nine classes of higher-degree binomial Painlevé equations, denoted BP-VII,..., BP-XV, of which the first seven are new. Two of these equations are of the third degree, two of the fourth degree, three of the sixth degree, and two of arbitrary degree n. All equations are solved in terms of the first, second or fourth Painlevé transcendents, elliptic functions, or quadratures. In the appendices, we discuss certain closely related classes of second-order nth equations (not necessarily of Painlevé type) which can also be solved in terms of Painlevé transcendents or elliptic functions.