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.     

Auto-Bäcklund transformations and special integrals for differential-delay Painlevé hierarchies

The six Painlevé equations have attracted much interest over the last thirty years or so. More recently many authors have begun to explore properties of higher-order versions of both these equations and their discrete analogues. However, little attention has been paid to differential-delay Painlevé equations, i.e., analogues of the Painlevé equations involving both shifts in and derivatives with respect to the independent variable, and even less to higher-order analogues of these last. In the current paper we discuss the phenomenon whereby members of one differential-delay Painlevé hierarchy define solutions of higher-order members of a second differential-delay Painlevé hierarchy. We also give an auto-Bäcklund transformation for a differential-delay Painlevé hierarchy. The key to our approach is the underlying Hamiltonian structure of related completely integrable lattice hierarchies.     

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.     

Generalization of the simplest equation method for nonlinear non-autonomous differential equations

It is known that the simplest equation method is applied for finding exact solutions of autonomous nonlinear differential equations. In this paper we extend this method for finding exact solutions of non-autonomous nonlinear differential equations (DEs). We applied the generalized approach to look for exact special solutions of three Painlevé equations. As ODE of lower order than Painlevé equations the Riccati equation is taken. The obtained exact special solutions are expressed in terms of the special functions defined by linear ODEs of the second order.     

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.     

Special solutions of the first and second Painlevé equations and singularities of the monodromy data manifold

A classification of solutions of the first and second Painlevé equations corresponding to a special distribution of poles at infinity is considered. The relation between this distribution and singularities of the two-dimensional complex monodromy data manifold used for the parameterization of the solutions is analyzed. It turns out that solutions of the Painlevé equations have no poles in a certain critical sector of the complex plane if and only if their monodromy data lie in the singularity submanifold. Such solutions belong to the so-called class of “truncated” solutions (intégrales tronquée) according to P. Boutroux’s classification. It is shown that all known special solutions of the first and second Painlevé equations belong to this class.     

The numerical solution of the second Painlevé equation

The Painlevé equations were discovered by Painlevé, Gambier and their colleagues during studying a nonlinear second‐order ordinary differential equation. The six equations which bear Painlevé's name are irreducible in the sense that their general solutions cannot be expressed in terms of known functions. Painlevé has derived these equations on the sole requirement that their solutions should be free from movable singularities. Many situations in mathematical physics reduce ultimately to Painlevé equations: applications including statistical mechanics, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics, and fiber optics. This fact has caused a significant interest to the study of these equations in recent years. In this study, the solution of the second Painlevé equation is investigated by means of Adomian decomposition method, homotopy perturbation method, and Legendre tau method. Then a numerical evaluation and comparison with the results obtained by the method of continuous analytic continuation are included. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On the nesting of Painlevé hierarchies: A Hamiltonian approach

We consider the phenomenon whereby two different Painlevé hierarchies, related to the same hierarchy of completely integrable equations, are such that solutions of one member of one of the Painlevé hierarchies are also solutions of a higher-order member of the other Painlevé hierarchy. An explanation is given in terms of the Hamiltonian structures of the related underlying completely integrable hierarchies, and is sufficiently generally formulated so as to be applicable equally to both continuous and discrete Painlevé hierarchies. Special integrals of a further Painlevé hierarchy related by Bäcklund transformation to the other Painlevé hierarchy mentioned above can also be constructed. Examples of the application of this approach to Painlevé hierarchies related to the Korteweg–de Vries, dispersive water wave, Toda and Volterra integrable hierarchies are considered. Our results provide further evidence of the importance of the underlying structures of related completely integrable hierarchies in understanding the properties of Painlevé hierarchies.     

The Painlevé Equations and Orthogonal Polynomials

This article is a survey of the recent studies jointly with Lies Boelen, Christophe Smet, Walter Van Assche and Lun Zhang (KULeuven, Belgium) on semi-classical continuous and discrete orthogonal polynomials and, in particular, on the connection of their recurrence coefficients to the solutions of the Painlevé equations. After recalling some basic facts about the Painlevé equations, we discuss continuous and discrete orthogonal polynomials and explain their connection.     

Some Algebraic Approach for the Second Painlevé Equation Using the Optimal Homotopy Asymptotic Method (OHAM)

The study of Painlevé equations has increased during the last years, due to the awareness that these equations and their solutions can accomplish good results both in the field of pure mathematics and in theoretical physics. In this paper we introduced the optimal homotopy asymptotic method (OHAM) approach to propose analytic approximate solutions to the second Painlevé equation. The advantage of this method is that it provides a simple algebraic expression that can be used for further developments while maintaining good performance and fitting closely the numerical solution.     

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.     

Symmetries of Quantum Lax Equations for the Painlevé Equations

Based on the fact that the Painlevé equations can be written as Hamiltonian systems with affine Weyl group symmetries, a canonical quantization of the Painlevé equations preserving such symmetries has been studied recently. On the other hand, since the Painlevé equations can also be described as isomonodromic deformations of certain second-order linear differential equations, a quantization of such Lax formalism is also a natural problem. In this paper, we introduce a canonical quantization of Lax equations for the Painlevé equations and study their symmetries. We also show that our quantum Lax equations are derived from Virasoro conformal field theory.     

Structural theory of special functions

A block diagram is suggested for classifying differential equations whose solutions are special functions of mathematical physics. Three classes of these equations are identified: the hypergeometric, Heun, and Painlevé classes. The constituent types of equations are listed for each class. The confluence processes that transform one type into another are described. The interrelations between the equations belonging to different classes are indicated. For example, the Painlevé-class equations are equations of classical motion for Hamiltonians corresponding to Heun-class equations, and linearizing the Painlevé-class equations leads to hypergeometric-class equations. The “confluence principle” is stated, and an example of its application is given. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 3–19, April, 1999.     

Euler integral symmetries for the confluent Heun equation and symmetries of the Painlevé equation PV

Euler integral symmetries relate solutions of ordinary linear differential equations and generate integral representations of the solutions in several cases or relations between solutions of constrained equations. These relations lead to the corresponding symmetries of the monodromy matrices for the differential equations. We discuss Euler symmetries in the case of the deformed confluent Heun equation, which is in turn related to the Painlevé equation PV. The existence of symmetries of the linear equations leads to the corresponding symmetries of the Painlevé equation of the Okamoto type. The choice of the system of linear equations that reduces to the deformed confluent Heun equation is the starting point for the constructions. The basic technical problem is to choose the bijective relation between the system parameters and the parameters of the deformed confluent Heun equation. The solution of this problem is quite large, and we use the algebraic computing system Maple for this.     

Nonclassical Symmetries and the Singular Manifold Method: Theory and Six Examples

In this paper we discuss a new approach to the relationship between integrability and symmetries of a nonlinear partial differential equation. The method is based heavily on ideas using both the Painlevé property and the singular manifold analysis, which is of outstanding importance in understanding the concept of integrability of a given partial differential equation. In our examples we show that the solutions of the singular manifold possess Lie point symmetries that correspond precisely to the so-called nonclassical symmetries. We also point out the connection between the singular manifold method and the direct method of Clarkson and Kruskal. Here the singular manifold is a function of its reduced variable. Although the Painlevé property plays an essential role in our approach, our method also holds for equations exhibiting only the conditional Painlevé property. We offer six full examples of how our method works for the six equations, which we believe cover all possible cases.     

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.     

Affine Weyl group symmetries of Frobenius Painlevé equations

In this paper, we introduce a Frobenius Painlevé IV equation and the corresponding Hamilton system, and we give the symmetric form of the Frobenius Painlevé IV equation. Then, we construct the Lax pair of the Frobenius Painlevé IV equation. Furthermore, we recall the Frobenius modified KP hierarchy and the Frobenius KP hierarchy by bilinear equations, then we show how to get Frobenius Painlevé IV equation from the Frobenius modified KP hierarchy. In order to study the different aspects of the Frobenius Painlevé IV equation, we give the similarity reduction and affine Weyl group symmetry of the equation. Similarly, we introduce a Frobenius Painlevé II equation and show the connection between the Frobenius modified KP hierarchy and the Frobenius Painlevé II equation.     

Numerical study of the asymptotics of the second Painlevé equation by a functional fitting method

The Painlevé equations arise as reductions of the soliton equations such as the Korteweg–de Vries equation, the nonlinear Schrödinger equation and so on. In this study, we are concerned with numerical approximation of the asymptotics of solutions of the second Painlevé equation on pole‐free intervals along the real axis. Classical integrators such as high order Runge–Kutta schemes might be expensive to simulate oscillation, decay and blow‐up behaviours depending on initial conditions. However, a lower order functional fitting method catches all kinds of solutions even for relatively large step sizes. Copyright © 2013 John Wiley & Sons, Ltd.     

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     

Integrable Nonautonomous Liénard-Type Equations

We study a family of nonautonomous generalized Liénard-type equations. We consider the equivalence problem via the generalized Sundman transformations between this family of equations and type-I Painlevé–Gambier equations. As a result, we find four criteria of equivalence, which give four integrable families of Liénard-type equations. We demonstrate that these criteria can be used to construct general traveling-wave and stationary solutions of certain classes of diffusion–convection equations. We also illustrate our results with several other examples of integrable nonautonomous Liénard-type equations.