The discrete and continuous Painlevé VI hierarchy and the Garnier systems

Abstract

Abstract. We present a general scheme to derive higher-order members of the Painleve´VI (P VI ) hierarchy of ODE's as well as their difference analogues. The derivation is based on a discrete structure that sits on the background of the P VI equation and that consists of a system of partial difference equations on a multidimensional lattice. The connection with the isomonodromic Garnier systems is discussed.1991 Mathematics Subject Classification. 34E99, 33E30, 58F07, 39A10.1. Introduction. In recent years there has been a growing interest in discrete analogues of the famous Painleve´equations; i.e. nonlinear nonautonomous ordinary difference equations tending to the continuous Painleve´equations in a well-defined limit and which are integrable in their own right; cf. In a recent paper [3] we established a connection between the continuous Painleve´VI (P VI ) equation and a non-autonomous ordinary difference equation depending on four arbitrary parameters. This novel example of a discrete Painleveé quation arises on the one hand as the nonlinear addition formula for the P VI transcendents, in fact what is effectively a superposition formula for its Ba¨cklund-Schlesinger transforms, on the other hand from the similarity reduction on the lattice (cf. [4,5]), of a system of partial difference equations associated with the lattice KdV family. In subsequent papers [6,7] some more results on these systems were established, namely the existence of the Miura chain and the discovery of a novel Schwarzian PDE generating the entire (Schwarzian) KdV hierarchy of nonlinear evolution equations and whose similarity reduction is exactly the P VI equation, this being to our knowledge the first example of an integrable scalar PDE that reduces to full P VI with arbitrary parameters.In the present note we extend these results to multidimensional systems associated with higher-order generalisations of the P VI equation. Already in [3] we noted that the similarity reduction of the lattice KdV system could be generalised in a natural way to higher-order differential and difference equations without, however, clarifying in detail the nature of such equations. We shall argue here that, in fact, such equations constitute what one could call the Painleve´VI hierarchy and its discrete counterpart. Whilst the idea of constructing hierarchies of Painleve´equations

A new two-dimensional lattice model that is consistent around a cube

“…How should integrability be defined for such maps? In [3] the following definition of integrability ("Consistency Around the Cube", CAC) was proposed (see Figure 2): Adjoin a third direction (therefore assuming x = x n,m,k ) and use the same map (but with different spectral parameters) also in planes corresponding to indices 1,3 and 2,3. That is, the map given in (1) contains shifts and parameters associated with directions 1,2 now the same should be done with directions 3,1 and 2,3, furthermore, on on the parallel shifted planes we use identical maps.…”

Cauchy problem for integrable discrete equations on quad-graphs

“…As it was observed in [9,10,11] there is a direct link between the Bäcklund transformation (7) and the discrete zero curvature (or Lax) representation for the equation (3). We will call the matrix L(x, y; α, λ) the Lax matrix for the equation (3) if this equation is equivalent to the relation…”

“…The notion of integrability laid in the basis of the classification in [12] is the 3-dimensional consistency [9,10,11] which means that the equation (3) may be consistently embedded into a 3-dimensional lattice, so that the similar equations hold for all six faces of any elementary cube, as on Fig. 2.…”

“…[8,9]). The general procedure [9,10,11] allows to construct for 3D consistent equations 2 × 2 Lax matrices in the following way. Put α = λ and rewrite the formula (7) in the form of the discrete Riccati equation…”

Рекурсионные Операторы, Законы Сохранения И Условия Интегрируемости Для Разностных Уравнений

“…П. ВАНГ, П. КСЕНИТИДИС некоторые методы и подходы из теории интегрируемых дифференциальных уравне-ний в частных производных (см., например, [13], [14]). В качестве критерия интегри-руемости на квадратной решетке (или квад-уравнений) для разностных уравнений было предложено [15], [16] свойство 3D-совместности, или совместности по сторо-нам куба [17]. Все интегрируемые аффинно-линейные квад-уравнения (удовлетво-ряющие некоторым дополнительным условиям симметрии относительно отражений решетки) были проклассифицированы Адлером, Бобенко и Сурисом (АБС) [18] (см.…”