Value distribution of Painlevé transcendents of the first and the second kind

We examine value distribution properties of the first and the second Painlevé transcendents. For every transcendental meromorphic solution ϕ(z) (resp. ψ(z)) of the first (resp. second) Painlevé equation, the deficiency δ(g,ϕ) (resp. δ(g, ψ)) of a small functiong(z) does not exceed 1/2. Furthermore, for ϕ(z), the ramification index satisfies ϑ(gφ)≤5/12.     

Poincaré series and monodromy of the simple and unimodal boundary singularities

A boundary singularity is a singularity of a function on a manifold with boundary. The simple and unimodal boundary singularities were classified by V.I. Arnold and V.I. Matov. The McKay correspondence can be generalized to the simple boundary singularities. We consider the monodromy of the simple, parabolic, and exceptional unimodal boundary singularities. We show that the characteristic polynomial of the monodromy is related to the Poincaré series of the coordinate algebra of the ambient singularity.     

Towards second order Lax pairs to discrete Painlevé equations of first degree

We investigate the question of finding discrete Lax pairs for the six discrete Painlevé equations (Pn). The choice we make is to discretize the pairs of Garnier, once converted to matrix form.     

Folding transformations of the Painlevé equations

New symmetries of the Painlevé differential equations, called folding transformations, are determined. These transformations are not birational but algebraic transformations of degree 2, 3, or 4. These are associated with quotients of the spaces of initial conditions of each Painlevé equation. We make the complete list of such transformations up to birational symmetries. We also discuss correspondences of special solutions of Painlevé equations.Acknowledgement The authors wish to thank Prof. Yosuke Ohyama, Prof. Shun Shimomura, and Dr. Yoshikatsu Sasaki for valuable discussions.     

Properties of solutions of two second-order differential equations with the Painlevé property

Theoretical and Mathematical Physics - We consider a Hamiltonian system equivalent to the Painlevé II equation with respect to one component and to the Painlevé XXXIV equation with...     

Deautonomisation of differential-difference equations and discrete Painlevé equations

We examine a family of integrable differential-difference equations and obtain their non-autonomous extensions using a discrete/continuous integrability criterion.     

Isomonodromic deformations of Heun and Painlevé equations

Continuing the study of the relationship between the Heun and the Painlevé classes of equations reported in two previous papers, we formulate and prove the main theorem expressing this relationship. We give a Hamiltonian interpretation of the isomonodromic deformation condition and propose an alternative classification of the Painlevé equations, which includes ten equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 3, pp. 395–406, June, 2000.     

Weak nonlinear asymptotic solutions for the fourth order analogue of the second Painlevé equation

The fourth-order analogue of the second Painlevé equation is considered. The monodromy manifold for a Lax pair associated with the P ₂ ² equation is constructed. The direct monodromy problem for the Lax pair is solved. Asymptotic solutions expressed via trigonometric functions in the Boutroux variables along the rays ? = \(\frac{2}{5}\)π(2n + 1) on the complex plane have been found by the isomonodromy deformations technique.     

Studies on the Painlevé equations

Summary In this series of papers, we study birational canonical transformations of the Painlevé system , that is, the Hamiltonian system associated with the Painlevé differential equations. We consider also -function related to and particular solutions of . The present article concerns the sixth Painlevé equation. By giving the explicit forms of the canonical transformations of associated with the affine transformations of the space of parameters of , we obtain the non-linear representation: GG_*, of the affine Weyl group of the exceptional root system of the type F₄ A canonical transformation of G_* can extend to the correspondence of the -functions related to . We show the certain sequence of -functions satisfies the equation of the Toda lattice. Solutions of , which can be written by the use of the hypergeometric functions, are studied in details.     

Invariance analysis and reduction of discrete Painlevé equations

The construction and role of symmetries for difference equations have been established, relatively, recently. In this paper, a symmetry analysis and reductions of the discrete Painlevé equations are considered. We assume that the characteristics of the ‘vector fields’ have a particular dependence since the general form lead to cumbersome calculations. Where possible, these symmetries are used to construct exact solutions in some cases.     

On deformations of linear systems of differential equations and the Painlevé property

We consider systems of linear differential equations discussing some classical and modern results in the Riemann problem, isomonodromic deformations, and other related topics. Against this background, we illustrate the relations between such phenomena as the integrability, the isomonodromy, and the Painlevé property. The recent advances in the theory of isomonodromic deformations presented show perfect agreement with that approach.     

On the Painlevé integrability,periodic wave solutions and soliton solutions of generalized coupled higher-order nonlinear Schrödinger equations

It is proven that generalized coupled higher-order nonlinear Schrödinger equations possess the Painlevé property for two particular choices of parameters, using the Weiss–Tabor–Carnevale method and Kruskal’s simplification. Abundant families of periodic wave solutions are obtained by using the Jacobi elliptic function expansion method with the assistance of symbolic manipulation system, Maple. It is also shown that these solutions exactly degenerate to bright soliton, dark soliton and mixed dark and bright soliton solutions with physical interests.     

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.     

Transport solutions of the Lamé equations and shock elastic waves

The Lamé system describing the dynamics of an isotropic elastic medium affected by a steady transport load moving at subsonic, transonic, or supersonic speed is considered. Its fundamental and generalized solutions in a moving frame of reference tied to the transport load are analyzed. Shock waves arising in the medium at supersonic speeds are studied. Conditions on the jump in the stress, displacement rate, and energy across the shock front are obtained using distribution theory. Numerical results concerning the dynamics of an elastic medium influenced by concentrated transport loads moving at sub-, tran- and supersonic speeds 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.     

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.