Nonlinear nonautonomous discrete dynamical systems from a general discrete isomonodromy problem

Nonlinear nonautonomous discrete dynamical systems (DDS) whose continuum limits are the well-known Painlevé equations, have recently arisen in models of quantum gravity. The Painlevé equations are believed integrable because each is the isomonodromy condition for an associated linear differential equation. However, not every DDS with an integrable continuum limit is necessarily integrable. Which of the many discrete versions of the Painlevé equations inherit their integrability is not known. How to derive all their integrable discrete versions is also not known. We provide a systematic method of attacking these questions by giving a general discrete isomonodromy problem. Discrete versions of the first and second Painlevé equations are deduced from this general problem.     

Integrability analysis of a conformal equation in relativity

In 1987 C. C. Dyer, G. C. McVittie, and L. M. Oattes derived the (two) field equations for shear-free, spherically symmetric perfect fluid spacetimes which admit a conformai symmetry. We use the techniques of the Lie and Painlevé analyses of differential equations to find solutions of these equations. The concept of a pseudo-partial Painlevé property is introduced for the first time which could assist in finding solutions to equations that do not possess the Painlevé property. The pseudo-partial Painlevé property throws light on the distinction between the classes of solutions found independently by P. Havas and M. Wyman. We find a solution for all values of a particular parameter for the first field equation and link it to the solution of the second equation. We indicate why we believe that the first field equation cannot be solved in general. Both techniques produce similar results and demonstrate the close relationship between the Lie and Painlevé analyses. We also show that both of the field equations of Dyeret al. may be reduced to the same Emden-Fowler equation of index two.     

Invertible point transformations,Painlevé test,and the second Painlevé transcendent

The techniques of invertible point transformations and the Painlevé analysis can be used to construct integrable ordinary differential equations. We compare both techniques for the second Painlevé transcendent.     

Bäcklund transformations for a matrix second Painlevé equation

Higher-order Painlevé equations are a topic of much current interest. Here we are interested in deriving auto-Bäcklund transformations for one particular kind of higher-order Painlevé equation, namely, a matrix Painlevé equation. The extension of a recently derived approach to deal with the matrix second Painlevé equation considered here represents a further demonstration of that approach's efficacy.     

Lax pair representation and Darboux transformation of noncommutative Painlevé’s second equation

Extension of the Painlevé equations to noncommutative spaces has been extensively investigated in the theory of integrable systems. An interesting topic is the exploration of some remarkable aspects of these equations, such as the Painlevé property, the Lax representation and the Darboux transformation, and their connection to well-known integrable equations. This paper addresses the Lax formulation, the Darboux transformation and a quasideterminant solution of the noncommutative form of Painlevé’s second equation introduced by Retakh and Rubtsov [V. Retakh, V. Rubtsov, Noncommutative Toda chain, Hankel quasideterminants and Painlevé II equation, J. Phys. A Math. 43 (2010) 505204].     

On the solvability of Painlevé II and IV

We introduce a rigorous methodology for studying the Riemann-Hilbert problems associated with certain integrable nonlinear ordinary differential equations. For concreteness we investigate the Painlevé II and Painlevé IV equations. We show that the Cauchy problems for these equations admit in general global, meromorphic int solutions. Furthermore, for special relations among the monodromy data and fort on Stokes lines, these solutions are bounded for finitet.     

Consistent Composition of Bäcklund Transformations Produces Confined Maps

We prove that difference equations arising from consistent compositions of Bäcklund transformations of the continuous Painlevé equations possess the singularity confinement property.     

The generalized Yablonskii-Vorob'ev polynomials and their properties

Rational solutions of the generalized second Painlevé hierarchy are classified. Representation of the rational solutions in terms of special polynomials, the generalized Yablonskii-Vorob'ev polynomials, is introduced. Differential-difference relations satisfied by the polynomials are found. Hierarchies of differential equations related to the generalized second Painlevé hierarchy are derived. One of these hierarchies is a sequence of differential equations satisfied by the generalized Yablonskii-Vorob'ev polynomials.     

Painlevé analysis,conservation laws,and symmetry of perturbed nonlinear equations

We consider the Lie-Backlund symmetries and conservation laws of a perturbed KdV equation and NLS equation. The arbitrary coefficients of the perturbing terms can be related to the condition of existence of nontrivial LB symmetry generator. When the perturbed KdV equation is subjected to Painlevé analysisa la Weiss, it is found that the resonance position changes compared to the unperturbed one. We prove the compatibility of the overdetermined set of equations obtained at the different stages of recursion relations, at least for one branch. All other branches are also indicated and difficulties associated them are discussed considering the perturbation parameter to be small. We determine the Lax pair for the aforesaid branch through the use of Schwarzian derivative. For the perturbed NLS equation we determine the conservation laws following the approach of Chen and Liu. From the recurrence of these conservation laws a Lax pair is constructed. But the Painlevé analysis does not produce a positive answer for the perturbed NLS equation. So here we have two contrasting examples of perturbed nonlinear equations: one passes the Painlevé test and its Lax pair can be found from the analysis itself, but the other equation does not meet the criterion of the Painlevé test, though its Lax pair is found in another way.     

On the Critical Behavior, the Connection Problem and the Elliptic Representation of a Painlevé VI Equation

In this paper we find a class of solutions of the sixth Painlevé equation appearing in the theory of WDVV equations. This class covers almost all the monodromy data associated to the equation, except one point in the space of the data. We describe the critical behavior close to the critical points in terms of two parameters and we find the relation among the parameters at the different critical points (connection problem). We also study the critical behavior of Painlevé transcendents in the elliptic representation.     

The Painlevé Property for Quasihomogenous Systems and a Many-Body Problem in the Plane

We investigate a many-body problem in the plane introduced by Calogero and intensively studied by Calogero, Françoise and Sommacal. An ad hoc complexification transforms the many-body problem to a system of second order autonomous complex equations depending on some complex constants that describe the two-body interactions. We investigate the sets of two-body interaction constants that make the complexified equation have the Painlevé Property, this is, its solutions are given by single-valued meromorphic functions. In this case the original system has only periodic isochronous solutions. We exhibit a family of settings where the system displays this property and show that it is not present in the three- and four-body problems that do not fall within our class. For this, we introduce a necessary condition for the presence of the Painlevé Property in some quasihomogenous systems.     

Singular manifold analysis of the Einstein vacuum field equations

The Painlevé property of the vacuum Einstein field equations is investigated. It is observed that the field equations possess this property when spacetime admits commuting, nonnull two Killing vector fields.     

Complete integrability of coupled KdV and NLS equations

Painlevé analysis is performed for the coupled system of nonlinear partial differential equations consisting of the KdV equation and NLS equation initially studied by Nishikawa. Various possibilities for the constants occurring in the system are explored, paying attention to the integrability property. This equation occurring in the field of plasma physics satisfies all the requirements of Painlevé analysis and can be ascertained to be completely integrable, though no Lax pair is known for it.     

Discrete Painlevé equations and their appearance in quantum gravity

We discuss an algorithmic approach for both deriving discrete analogues of Painlevé equations as well as using such equations to characterize similarity reductions of spatially discrete integrable evolution equations. As a concrete example we show that a discrete analogue of Painlevé I can be used to characterize similarity solutions of the Kac-Moerbeke equation. It turns out that these similarity solutions also satisfy a special case of Painlevé IV equation. In addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions. As an example we use the WKB method in the presence of two turning points of the third order to parametrize (at the continuous limit) the solution of Painlevé I in terms of the solution of discrete Painlevé I. Finally we show that these results are useful for investigating the partition function of the matrix model in 2D quantum gravity associated with the measure exp [–t ₁ z ² –t ₂ z ⁴ –t ₃ z ⁶].     

Similarity solution and lie symmetry for a coupled nonlinear system

The Lie point symmetries of a set of coupled nonlinear partial differential equations are considered. The system is an extended version of the usual nonlinear Schrödinger equation. In the similarity variable deduced from the symmetry analysis, the system is equivalent to the Painlevé III in Ince's classification. By starting from a solution of the Painlevé equation, one can reproduce various classes of solutions of the original PDEs. Such solutions include both rational and progressive types or a combination of the two.     

On the Painlevé classification of partial differential equations II. Rational equations with one pole

The singular point analysis and the finite point analysis is used for the Painlevé classification of semilinear second order partial differential equations in two independent variables.     

Prolongation structure and Painlevé property of the Gürses-Nutku equations

It is shown that the Gürses-Nutku equations have a finite prolongation algebra for any value of the parameterK. The Painlevé property of these equations is also examined.The research reported in this paper is supported in part by the Scientific and Technical Research Council of Turkey.     

Nonlinear wave equations,covariant exterior derivative,Painlevé test,and integrability

A class of nonlinear wave equations is derived which can be written as the vanishing of a covariant exterior derivative. The Painlevé test is performed and the connection with integrability is discussed.