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.

The triply truncated solutions of the first Painlevé equation were specified by Boutroux in his famous paper of 1913 as those having no poles (of large modulus) except in one sector of angle 2π/5. There are five such solutions and each of them can be obtained from any other one by applying a certain symmetry transformation. One of these solutions is real on the real axis. We found a characteristic property of this solution, different from the asymptotic description given by Boutroux. This allows us to estimate numerically the position of its real pole and zero closest to the origin. We also study properties of asymptotic series for truncated solutions.

We present a novel integrable non-autonomous partial differential equation of the Schwarzian type, i.e. invariant under Möbius transformations, that is related to the Korteweg-de Vries hierarchy. In fact, this PDE can be considered as the generating equation for the entire hierarchy of Schwarzian KdV equations. We present its Lax pair, establish its connection with the SKdV hierarchy, its Miura relations to similar generating PDE's for the modified and regular KdV hierarchies and its Lagrangian structure. Finally we demonstrate that its similarity reductions lead to the full Painlevé VI equation, i.e.

No abstract

Abstract.The second Painlevé hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well-known second Painlevé equation, P II .In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlevé analysis to ordinary differential equations. We extend these techniques in order to derive autoBäcklund transformations for the second Painlevé hierarchy. We also derive a number of other Bäcklund transformations, including a Bäcklund transformation onto a hierarchy of P 34 equations, and a little known Bäcklund transformation for P II itself.We then use our results on Bäcklund transformations to obtain, for each member of the P II hierarchy, a sequence of special integrals.

No abstract

We present a generalized non-isospectral dispersive water wave hierarchy in 2 + 1 dimensions. We characterize our entire hierarchy and its underlying linear problem using a single equation together with its corresponding non-isospectral scattering problem. This then allows a straightforward construction of linear problems for the entire generalized 2 + 1 hierarchy. Reductions of this hierarchy then yield new integrable hierarchies in 1 + 1 dimensions, and also new integrable hierarchies of ordinary differential equations, all together with their underlying linear problems. In particular, we obtain a generalized PIV − PII hierarchy; this includes as special cases both a hierarchy of ODEs having the fourth Painlevé equation as first member, and also a hierarchy of ODEs having the second Painlevé equation as first member. All of these hierarchies of ordinary differential equations, as well as their underlying linear problems, are new; both the PIV hierarchy and the PII hierarchy obtained here are different from those which have previously been given.

No abstract