Abstract,We review the different aspects of integrable discretizations in space and time of the Korteweg-de Vries equation, including Miura transformations to related integrable difference equations, connections to integrable mappings, similarity reductions and discrete versions of Painlev6 equations as well as connections to Volterra systems.Mathematics Subject Classification (1991): 58F07.

In the paper [V. Adler, IMRN 1 (1998) 1-4] a lattice version of the Krichever-Novikov equation was constructed. We present in this note its Lax pair and discuss its elliptic form.

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

This first introductory text to discrete integrable systems introduces key notions of integrability from the vantage point of discrete systems, also making connections with the continuous theory where relevant. While treating the material at an elementary level, the book also highlights many recent developments. Topics include: Darboux and Bäcklund transformations; difference equations and special functions; multidimensional consistency of integrable lattice equations; associated linear problems (Lax pairs); connections with Padé approximants and convergence algorithms; singularities and geometry; Hirota's bilinear formalism for lattices; intriguing properties of discrete Painlevé equations; and the novel theory of Lagrangian multiforms. The book builds the material in an organic way, emphasizing interconnections between the various approaches, while the exposition is mostly done through explicit computations on key examples. Written by respected experts in the field, the numerous exercises and the thorough list of references will benefit upper-level undergraduate, and beginning graduate students as well as researchers from other disciplines.

No abstract