Initial value problems for discrete bilinear equations there are many equations that. Spacetime discrete integrable systems with periodic. Integrable system an overview sciencedirect topics. Discrete integrable systems and cluster algebras 29 theorem 2. Reflection groups and discrete integrable systems journal. Discrete integrable systems have received a lot of attention in the last two decades. Of particular interest to us is the natural and profitable presence of cluster algebras within the field of discrete integrable systems 105. Invariant measures for kdv and todatype discrete integrable. He was the principal organizer of the 2009 sixmonth programme on discrete integrable systems at the isaac newton institute, and a royal society leverhulme trust senior research fellow in 2011.
Mkdv, sg, pkdv, schwarzian kdv, boussinesq and modi. Recent intensive studies revealed that mathematical structures of discrete integrable systems are richer than those of continuous integrable systems. It constitutes a companion book to integrability of nonlinear systems springerverlag, 2004, lnp 638, isbn 3540206302. Pdf darboux transformations, discrete integrable systems and.
Spectral theory of difference operators and discrete systems chapter ii. Areas of physics in which discrete integrable systems prominently feature include statistical mechanics and discrete analogues of integrable systems in classical mechanics or solid state physics 58, 2225. Consequently, the natural problem of extending the geometric interpretation of integrable partial differential equations to the discrete domain arose, leading not only to. For example, during the first semester of 2009, the isaac newton institute for mathematical sciences in cambridge u. Research article a hierarchy of discrete integrable. Mathematics, differential and integral equations, dynamical systems and control theory, physics and. We construct the directed graph quiver for which the associated cluster. However, geometric interpretation of most of the discrete integrable systems considered in the. Discrete and continuous coupled nonlinear integrable systems.
Indeed, the study of discrete systems has become one of the focal points in the integrable systems community. We show that all hamiltonians, partition functions of all weighted perfect matchings with a common homology class. We define infinite versions of four wellstudied discrete integrable models, namely the ultra discrete kdv equation, the discrete kdv equation, the ultra discrete toda equation, and the discrete toda equation. This volume consists of a set of ten lectures conceived as both introduction and uptodate survey on discrete integrable systems. Discrete integrable systems isaac newton institute. Laurent phenomenon of difference equations in this chapter we will show that many difference equations have laurentness. Pdf special solutions of discrete integrable systems. Pdf foreword to the special issue on continuous and. This observation is applied to define the dynamics uniquely on the infinite configuration space and study the invariant measures. Discrete versions of some classical integrable systems and. Spacetime discrete integrable systems with periodic boundary condition department of physics, school of science, kitasato university, kanagawa, 2288555 japan kiyoshi sogo february 16, 2007 1.
We find a poisson structure on the space of twisted polygons and show that the pentagram map relative to this poisson structure is completely integrable in the. Classical differential geometry discretization principles. Discrete integrable systems qrt maps and elliptic surfaces. Geometry of integrable lattice equations and their reductions. Unifying scheme for generating discrete integrable systems. 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. This viewpoint is adopted by many authors see 9, 14 and references therein and this is precisely our viewpoint in the discrete case. Moreover, the algebraic approach in classical integrable models, though has a rich and sophis ticated formulation through the classical yangbaxter equation and the classical rmatrix 5.
In this paper we study a class of discrete integrable systems which are closely related to problems occurring in mathematical physics such as the heisenberg. Cambridge core nonlinear science and fluid dynamics discrete systems and integrability. Discrete integrable systems 19 january to 3 july 2009. A twisted polygon is a map from z into the projective plane that is periodic modulo a projective transformation. We consider a discrete dynamical system given by the dthorder o e. However, geometric interpretation of most of the discrete integrable systems considered in the mathematical and physical literature is unclear. By comparison, discrete integrable systems will be introduced when they.
According to the concept of multidimensional consistency, integrable twodimensional systems can be imposed in a consistent way on all twodimensional sublattices of a lattice of arbitrary dimension. While this school focuses on discrete integrable systems we feel it nec. Discrete integrable systems, matrix orthogonal polynomials and. Zabrodin february 1, 2008 abstract functional relation for commuting quantum transfer matrices of quantum integrable models is identi. Discrete integrable systems also yield some new phenomena.
It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. It is well known that the theory of continuous integrable system has. Discrete integrable systems qrt maps and elliptic surfaces pdf here. Cluster algebras and discrete integrable systems ideals. In mathematics, integrability is a property of certain dynamical systems. What is integrability of discrete variational systems. On orthogonal polynomials and related discrete integrable. General solutions for kdv and todatype discrete integrable. This secondary introduction, will give a brief look at some of the key equations and systems studied in discrete integrable systems as well as an introduction into the painleve equations. Oct 31, 2008 the pentagram map is a projectively natural iteration defined on polygons, and also on objects we call twisted polygons a twisted polygon is a map from z into the projective plane that is periodic modulo a projective transformation. Discrete integrable systems and geometric numerical. These arise as analogues of curvature ows for polygon evolutions in homogeneous spaces, and this is the focus of the second half of the paper. While treating the material at an elementary level, the book also highlights many recent developments. It reflects the recent progress in discrete differential geometry and contains many original results.
Indeed, this was conjectured somewhat loosely in 21. In this paper, we construct the addition formulae for several integrable hierarchies, including the discrete kp, the qdeformed kp, the twocomponent bkp and the d type drinfeldsokolov hierarchies. On soliton solutions of multicomponent semidiscrete. In the geometrical setting which is more close to our viewpoint, see 5 for many interesting examples. As a rst step in this direction, the editorial board has the honor to o er you this special issue dedicated to continuous and discrete inte. Moreover, we give an introduction to the theory of yangbaxter maps and we show its relation to discrete integrable systems. Center manifold approach to discrete integrable systems. Integrable circle patterns foundations solutions of selected exercises bibliography notations index. It constitutes a companion book to integrability of nonlinear system.
Consequently, the natural problem of extending the geometric interpretation of integrable partial differential equations to the discrete. Discrete integrable systems have also applications in many other fields such as numerical analysis, discrete geometry, mathematical biology, and economics. A discrete analog of the dressing method is presented and used to derive integrable nonlinear evolution equations, including two infinite families of novel continuous and discrete coupled integrable. Recent intensive studies revealed that mathematical structures of discrete. Continuous and discrete integrable systems with applications applicable analysis wishes to become a leading journal for highquality papers on all mathematical aspects of completely integrable systems. The short pulsespequation is an integrable equation. Sistemas integrables discretos, polinomios matriciales ortogonales. This is a development along the line of research of discrete integrable lagrangian systems initiated in 2009 by lobb and nijhoff, however, having its more. Editorial foreword to the special issue on continuous and.
On orthogonal polynomials and related discrete integrable systems. Selfadaptive moving mesh discrete integrable systems. Pdf discrete integrable systems and poisson algebras from. Quantum integrable systems and elliptic solutions of classical discrete nonlinear equations i. Discrete integrable systems isaac newton institute for. Laurent phenomenon of difference equations in this chapter we will show that many difference equations have laurentness by applying theorem 2. Bianchi permutability nonlinear superposition of backlund transformations leads directly to fully discrete equations. Intuitively, an integrable system is a di erential equation we can \integrate up directly. Discrete integrable systems and cluster algebras rims, kyoto. Discrete systems and integrability cambridge university press. Request pdf discrete integrable systems three lessons on the painleve property and the painleve equations m.
In this paper, we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of korteweg. Discrete integrable systems have played an increasingly prominent part in mathematical physics. We derive darboux matrices for all the above cases and we use them to construct novel discrete integrable systems together with their lax representations. Discrete integrable systems 19 january to 3 july 2009 final.
Integrable symplectic maps associated with discrete. On the other hand, discrete holomorphic quadratic differentials are closely related to discrete integrable systems. It contains the full texts pdf of all volumes published since. Discrete integrable systems, darboux transformations, and.
First appeared in 1970s but active area of research in last decade. The study of discrete integrable systems is rather new. Between continuous and discrete in integrable systems. The papers 1,2 considers a discrete integrable systems on the space of ngons, different from the pentagram map. Pdf discrete integrable systems and poisson algebras. Today, all the topics for continuous integrable systems have been.
They appear in the discrete equations of toda type 1, 4. These systems arise from the theory of discrete integrable systems. Integrable discrete systems on r and related dispersionless systems maciej blaszak1, metin g. Symmetries of zn graded discrete integrable systems. Nets in quadrics special classes of discrete surfaces approximation consistency as integrability discrete complex analysis. Indeed, in the \smooth case it is natural to consider the full space of linear di erential equations. Uccs math department colloquium series discrete integrable. For example, during the first semester of 2009, 1 the isaac newton institute for mathematical sciences in cambridge u. Feb 08, 2014 this led to classification of discrete twodimensional integrable systems abs list, which turned out to be rather influential. With the help of the hirota bilinear equations and. While in theory, integrable systems should be very rare, it happens that in nature, a lot of systems happen to be integrable. Zn graded discrete lax pairs and discrete integrable systems.
By means of the discrete quadraticform identity we establish the hamiltonian structures of the hierarchy. On the factorizations of rational matrix functions with. This concept has arisen from the study of cluster algebras, for which it is known that any. Discrete integrable systems is a new and actively developing subject, hundreds of new articles in this. The multidimensional consistency property of the discrete integrable system is shown to be inherited from the combinatorial properties of the polytope. Discrete integrable systems and geometric numerical integration. The writers of discrete integrable systems qrt maps and elliptic surfaces have made all reasonable attempts to offer latest and precise information and facts for the readers of this publication. Introduction the theory of integrable dynamical systems has an unexpected relationship to numerical algorithms.
The recent progress in studying discrete integrable systems showed that, in many respects, they should be considered as more fundamental than their differential counterparts. The laurent phenomenon and discrete integrable systems 47 shown the laurentness of several discrete equations 1, among which several famous discrete integrable systems, for example the discrete \mathrmk\mathrmd\mathrmv equation, the hirota. Discrete integrable systems introduction conservation laws of lattice kdv the lattice kdv family full lattice kdv hdkdv in fact an integrable quadgraph equation. Some integrable systems of lax form describe continuous. It began with discretising continuous integrable systems in 1970s. Discrete and continuous coupled nonlinear integrable. Foreword to the special issue on continuous and discrete. Manin involutions for elliptic pencils and discrete. Selfadaptive moving mesh discrete integrable systems kenichi maruno department of mathematics, the university of texas pan american joint work with baofeng feng utpa, kenji kajiwara kyushu university, yasuhiro ohta kobe university conference on nonlinear mathematical physics, sophus lie conference center, norway june 414, 20. Zn graded discrete lax pairs and discrete integrablesystems introduction and outline integrable discretisations of soliton equations. An extension of the notion of integrability is also applicable to discrete systems such as lattices. Discrete integrable systems 19 january to 3 july 2009 final report organisers.
Discrete analogues of classical integrable systems and factorization of matrix polynomials 3. The box ball system, discrete integrable systems and. On discrete integrable systems hilaris publishing srl. We explain the role of darboux and backlund transformations in the theory of integrable systems, and we show how they can be used to construct discrete integrable systems via the laxdarboux scheme. We study a discrete dynamic on weighted bipartite graphs on a torus, analogous to dimer integrable systems of goncharov and kenyon 20. For each equation, we show that there exists a unique solution to the initial value problem when the given data. This thesis deals with discrete integrable systems theory and modified hamiltonian equa tions in the field of geometric numerical integration. The laurent phenomenon and discrete integrable systems by takafumi mase abstract the laurent phenomenon is the property that the solution to an initial value problem of a discrete equation is expressed as a laurent polynomial of the initial values. In the following, we focus on examples from discrete holomorphic maps in nonlinear discrete complex analysis, which are known to possess discrete integrable structures. Discrete integrable systems and poisson algebras from cluster. Pdf discrete moving frames and discrete integrable systems. N graded discrete integrable systems allan p fordy1 and pavlos xenitidis2,3 1 schoolofmathematics,universityofleeds,leedsls29jt,unitedkingdom 2 schoolofmathematics. Both dkdv and dtoda can be obtained from the discrete kp equation.
On the other hand, it is addressed to specialists in geometry and mathematical physics. Discrete integrable systems basil grammaticos springer. A number of intriguing connections have emerged between the field of discrete integrable systems and various areas of mathematics and physics in the past two decades 1, 2. A geometric interpretation of all the discrete integrable systems considered in the above references is unclear. Further, the resulting hamiltonian equations are all proved to be integrable in liouville sense.
130 1326 612 576 488 1350 113 967 979 698 390 469 90 1210 268 991 405 1354 1051 908 210 961 267 1078 241 459 356 1400 1074 209 1470