Hierarchies of Difference Equations and Bäcklund Transformations (original) (raw)

Bäcklund transformations for discrete Painlevé equations : dP II-dP

2004

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, dPIIdPV, with different parameters. Rational solutions and elementary solutions of discrete Painlevé equations can also be obtained from these transformations.

On the relation between the continuous and discrete Painlev� equations

Theor Math Phys Engl Tr, 2000

In this paper we discuss a method for deriving difference equations, in particular the discrete Painlevé equations, from the Bäcklund transformations of the continuous Painlevé equations. Using this technique one is able to derive several of the known discrete Painlevé equations, in particular the first and second discrete Painlevé equations and some of their alternative versions. It is well-known that 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 using the aforementioned relationships, hierarchies of exact solutions for the associated discrete Painlevé equations can be generated. Thus exact solutions of the Painlevé equations simultaneously satisfy both a differential equation and a difference equation, in analogy with the special functions.

Bäcklund transformations for discrete Painlevé equations: Discrete PII–PV

Chaos, Solitons & Fractals, 2006

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.

Bäcklund Transformations for First and Second Painlevé Hierarchies

2009

We give B\"acklund transformations for first and second Painlev\'e hierarchies. These B\"acklund transformations are generalization of known B\"acklund transformations of the first and second Painlev\'e equations and they relate the considered hierarchies to new hierarchies of Painlev\'e-type equations.

On the alternate discrete Painlevé equations and related systems

Séminaires et Congres de …, 2006

We examine the family of discrete Painlevé equations which were introduced under the qualifier of "alternate". We show that there exists a transformation between the two canonical forms of these equations, and we proceed to link these forms to the contiguity relations of the continuous P VI. We describe the full degeneration cascade of this contiguity, obtaining all related discrete Painlevé equations (among which, one which has never been derived before) as well as mappings which are solvable by linearisation. Résumé (Sur les équations discrètes alternatives de Painlevé et les systèmes associés) Nousétudions la famille deséquations de Painlevé discrètes dites « alternatives ». Nous exhibons une transformation entre les deux formes canoniques et nous relions celles-ci aux relations de contiguïté de l'équation de Painlevé continue P VI. Nous décrivons la cascade de dégénérescence complète liéeà cette contiguïté ; nous explicitons toutes leséquations de Painlevé discrètes correspondantes (dont une inconnueà ce jour) ainsi que des applications résolubles par linéarisation.

Bäcklund transformations for higher order Painlevé equations

Chaos, Solitons & Fractals, 2004

We present a new generalized algorithm which allows the construction of Bäcklund transformations (BTs) for higher order ordinary differential equations (ODEs). This algorithm is based on the idea of seeking transformations that preserve the Painlevé property, and is applied here to ODEs of various orders in order to recover, amongst others, their auto-BTs. Of the ODEs considered here, one is seen to be of particular interest because it allows us to show that auto-BTs can be obtained in various ways, i.e. not only by using the severest of the possible restrictions of our algorithm.

On the extension of the Painlevé property to difference equations

2000

It is well known that the integrability (solvability) of a differential equation is related to the singularity structure of its solutions in the complex domain-an observation that lies behind the Painlevé test. A number of ways of extending this philosophy to discrete equations are explored. First, following the classical work of Julia, Birkhoff and others, a natural interpretation of these equations in the complex domain as difference or delay equations is described and it is noted that arbitrary periodic functions play an analogous role for difference equations to that played by arbitrary constants in the solution of differential equations. These periodic functions can produce spurious branching in solutions and are factored out of the analysis which concentrates on branching from other sources. Second, examples and theorems from the theory of difference equations are presented which show that, modulo these periodic functions, solutions of a large class of difference equations are meromorphic, regardless of their integrability. It is argued that the integrability of many difference equations is related to the structure of their solutions at infinity in the complex plane and that Nevanlinna theory provides many of the concepts necessary to detect integrability in a large class of equations. A perturbative method is then constructed and used to develop series in z and the derivative of log (z), where z is the independent variable of the difference equation. This method provides an analogue of the series developed in the Painlevé test for differential equations. Finally, the implications of these observations are discussed for two tests which have been studied in the literature regarding the integrability of discrete equations.

Transformations of Painleve Equations

2005

In this article, we studied the usage of the one-to-one correspondence between a given Painlevé equation and a certain second-order seconddegree Painlevé-type equation to derive the Bäcklund transformations for the given Painlevé equation. We showed that all basic Bäcklund transformations of the second, third, and fourth Painlevé equations can be derived by using the one-to-one correspondences between these equations and the Cosgrove's SD-I equation.

Bäcklund Transformations of Some Nonlinear Evolution Equations VIA Painlevè Analysis

Abstract: In this paper we present the Painlevè test for the (1+1) –dimensional travelling regularized long wave (TRLW) equation, the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation, the modified improved Kadomtsev-Petviashvili equation (MIKP) and the variant shallow water wave equations. The associated Bäcklund transformations are obtained directly from the Painlevè test. Keywords: the (1+1) –dimensional travelling regularized long wave (TRLW) equation, the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation, the modified improved Kadomtsev-Petviashvili equation (MIKP), the variant shallow water wave equations and Painlevè analysis.