Lorentz Center - Discrete Integrable Systems from 18 Jul 2011 through 22 Jul 2011
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    Discrete Integrable Systems
    from 18 Jul 2011 through 22 Jul 2011

 

Abstracts

 

Singular-boundary reductions of Q4

James Atkinson

 

A new type of reduction and its explicit solution will be presented for the well-known Q4 lattice equation. The reduction results from imposing a prescribed singularity configuration globally on the lattice in such a way that there remain only a finite number of degrees of freedom. A method of solving the reduced system is obtained from the principal integrability feature of multidimensional consistency. Important auxiliary roles are played by the singularity structure in multi-dimensions and a natural simplification of the problem on the level of an associated tau-function. How these solutions fit into the broader picture will be discussed by explaining their relationship with Soliton and periodic solutions, and by giving the associated solutions of the lattice-KP equation.

 

 

Discrete integrable systems with convex variational principles

Alexander Bobenko

 

Two-dimensional integrable systems on quad-graphs with convex variational principles are investigated. The convexity of the action functional implies the uniqueness of the solution of the Dirichlet boundary value problem. The solution can be obtained by minimization of the functional. Applications to discrete conformal mappings are discussed.

 

 

On Lagrangian Structures of 3D Consistent Systems of Quad-Equations

Raphael Boll

 

One of the key features of discrete integrable systems which recently became the focus of attention is the flip invariance of action functionals in multidimensional lattices. This invariance was proven for systems of quad-equations from the ABSlist by Bobenko and Suris. Recently, we gave a classification of all 3D consistent 6-tuples of equations with the tetrahedron property, where several novel asymmetric systems have been found. We establish Lagrangian structures and flip invariance for asymmetric systems coming from this classification and for the corresponding Laplace type equations.

 

 

Invariants of non-QRT mappings and rational elliptic surfaces of higher index

Stefan Carstea

 

In this talk we discuss an extension of the Sakai algebraic-geometric approach to second order non-QRT mappings preserving elliptic fibrations. The interesting findings are related  to the invariants corresponding to Halphen surfaces of higher index.  The action of the mapping may or may not exchange their fibers. We  give a classification of such mappings which preserve an elliptic  fibration and finally three examples as explicit realizations of the  results. Their relations with Painleve equations is discussed as well.

 

 

Integrable and non-integrable dual equations

Dmitry Demskoi

 

The presentation is concerned with the question of integrability of dual equations generated by integrals of KdV-like equations. We demonstrate that, in general, the answer to this question is negative. We use the polynomial growth of degrees of iterates as indicator of (non)integrability. However, wide classes of integrable dual equations do exist. We give a few examples of such classes.

 

 

Pentagon equation, incidence geometry, and Hirota's discrete KP equation

Adam Doliwa

 

It is shown that the Desargues configuration provides a natural solution to functional dynamical pentagon equation with variables in an arbitrary division ring (skew field). In an appropriate gauge the commutative version of the map preserves a natural Poisson structure, which is the quasiclassical limit of the Weyl commutation relations. The corresponding solution of the quantum dynamical pentagon equation is given in terms of the noncompact quantum dilogarithm. The relation to noncommutative version of Hirota's discrete Kadomtsev--Petviashvili equation will be discussed.

 

This is joint work with Sergey M. Sergeev (University of Canberra, Australia).

 

 

Growth and Integrability

Rod Halburd

 

Starting from a class of second-order discrete equations, slow height growth of an admissible solution will be used to single out a discrete Painleve equation and some nonintegrable equations for which the solution also satisfies a discrete Riccati equation.  Results on differential-delay equations will be described.

 

 

Bilinearization and special solutions to the discrete Schwarzian KdV equation

Mike Hay

 

Motivated by the importance of the discrete Schwarzian KdV equation (dSKdV) for discrete geometry, where it can be considered to be a formula for the discrete conformality of maps, we construct various solutions for it in terms of τ functions.

Firstly, this is done by considering the discrete Schwarzian KP equation (dSKP) and establishing a reduction procedure to achieve the non-autonomous dSKdV. dSKdV appears not only as one of the discrete soliton equations, but also as the equation describing the chain of Backlund transformations of the Painleve systems. This implies that dSKdV also admits solutions expressible in terms of the τ functions of Painleve equations. We discuss dSKdV in the setting of the Painleve systems and construct explicit solutions in terms of their τ functions as well."

 

 

An explicit formula for the discrete power function associated with circle patterns of Schramm type

Kenji Kajiwara

 

We present an explicit formula for the discrete power function introduced by Bobenko, which is expressed in terms of the hypergeometric tau functions for the sixth Painlevé equation. This is a generalization of the result by Agafonov in 2005. As a byproduct, we show that one can extend the value of the exponent to arbitrary complex numbers except even integers and the domain to a discrete analogue of the Riemann surface.

 

 

Analytic theory of linear difference equations and its applications

Igor Krichever

 

The problem of convergence of solutions of difference equations to solutions of differential equations is central in many areas of mathematical physics. The talk will be devoted to the possible applications of analytic theory of difference equation for the study of convergence problem for discrete analogs of Painleve equations.

 

 

Symmetries and transformations for integrability and linearizability tests of quad--graph equation

Decio Levi

 

In this presentation we introduce at first, using the formal symmetry approach, an integrability test for partial difference equations on a square lattice. Then, in a similar fashion, we consider linearizability tests for the same kind of equations based on point, contact and Hopf-Cole transformations. We end this presentation by presenting a few applications.

 

 

Integrability of long-legged equations

Sarah Lobb

 

We consider the integrability of 3- and higher-dimensional discrete equations, stemming from relations between the long legs of the three leg form of equations in the Adler-Bobenko-Suris classification. The fundamental equations in this class are those involving 3 dimensions; for these we show multidimensional consistency, give a system of Lax matrices, and give closed Lagrangian forms.

 

 

Renormalization, isogenies and rational symmetries of differential equations

Jean-Marie Maillard

 

We give several examples of infinite order rational transformation that leaves  linear differential equations covariant. These examples can be seen as a non-trivial but still simple illustration of an exact representation of the renormalization group.

 

 

Discrete Integrable Systems and Hodograph Transformations Arising from Motions of Discrete Plane Curves

Kenichi Maruno

 

We present integrable discretizations of several soliton equations which have loop soliton and cusp soliton solutions. In this talk, the WKI loop soliton equation and the Dym equation are discussed in detail. In the continuous case, there is a link between either the mKdV equation and the WKI equation, or the mKdV equation and the Dym equation through hodograph transformations. An important task of integrable discretizations of these equations is how to handle hodograph transformations in discrete systems. We present a geometric method to make integrable discretizations of the WKI and Dym equations. Based on the formulation of motion of discrete plane curves, integrable discretizations of hodograph transformations are obtained in a systematic way. We discuss the motion of discrete plane curves on both Euclid plane (focusing case) and Minkowski plane (defocusing case). We also give explicit formulas of geometric quantities by tau-functions.

 

 

q-Schlesinger transformations for an associated linear problem for q-P(A_2^{(1)})

Christopher Ormerod

 

We present a new associated linear problem for the q-difference Painleve equation whose space of initial conditions is the A_2^{(1)} surface. We consider the group of connection preserving deformations for this associated linear problem and show that the Backlund transformations for this equation may expressed as connection preserving deformations. We will also discuss further developments in lifting the Backlund transformations to the level of the associated linear problems.

 

 

Yang-Baxter maps, dynamic YB maps, entwining YB maps and integrable  lattices

Vassilios Papageorgiou

 

We review 3d-compatible discrete systems from the point of view of Yang-Baxter (YB) maps. In particular we discuss how 3d compatible equations on vertices can be cast in the form of dynamical YB maps. We present also set-theoretic analoga of entwining Yang-Baxter  (YB) structures i.e. YB map systems with Lax triples L1, L2, L3 derived from symplectic leaves of 2 x 2 binomial matrices equipped with the Sklyanin bracket A unique factorization condition of the Lax triple implies a 3-dimensional compatibility property of these maps. In case of one Lax matrix (i.e. L1, L2, L3 are of the same form) this property yields the set-theoretic quantum Yang-Baxter equation, i.e. the YB map property. By considering periodic 'staircase' initial value problems on quadrilateral lattices, these maps give rise to multidimensional integrable mappings which preserve the spectrum of the corresponding monodromy matrix.

 

 

The relation between quantum relativistic Calogero-Moser systems and quantum Knizhnik-Zamolodchikov equations

Jasper Stokman

 

Both quantum relativistic Calogero-Moser (CM) systems and quantum Knizhnik-Zamolodchikov (KZ) equations are integrable structures defined in terms of $q$-difference operators. In this talk we show that the spectral analysis associated to the quantum  relativistic CM systems is equivalent to the analysis of the solutions of the quantum KZ equations.

 

 

Integrability of the Pentagram map

Fedor Soloviev

 

The Pentagram map was introduced by R. Schwartz in 1992 for convex planar polygons. Recently, V. Ovsienko, R. Schwartz, and S. Tabachnikov proved Liouville integrability of the pentagram map for generic monodromies by providing a Poisson structure and the sufficient number of integrals in involution on the space of twisted polygons. In this paper we prove algebraic-geometric integrability for any monodromy, i.e., for both twisted and closed polygons. For that purpose we show that the pentagram map can be written as a discrete zero-curvature equation with a spectral parameter, study the corresponding spectral curve, and the dynamics on its Jacobian. We also prove that on the symplectic leaves Poisson brackets discovered for twisted polygons coincide with the symplectic structure obtained from Krichever-Phong's universal formula.

 

 

Integrable Partial Difference Equations and Their Reductions

Reinout Quispel

 

Integrable ordinary differential equations (ODEs) have been studied intensively since the time of Kepler and Newton, integrable partial differential (PDEs) equations since the discovery of solitons in the Korteweg-de Vries equation by Kruskal and collaborators. Both integrable ODEs as well as PDEs are of great importance in mathematics as well as in applications (e.g. as a starting point for perturbation theory).

 

The question thus arises how to generalize this theory and its applications from the continuous case (integrable differential equations) to the discrete case (integrable difference equations), i.e. when all independent variables are discrete. This question is important both from the point of view of mathematics (where the discrete case may be viewed as a deformation of the continuous case), as well as from the point of view of applications. The history of discrete integrable systems goes back to seminal papers in the 1970s by Ablowitz and Ladik and by Hirota. In the early 1980s, systematic methods for the construction of integrable nonlinear partial difference equations (PΔEs) were found, e.g. through the representation theory of infinite-dimensional Lie algebra, or in the work of Quispel, Nijhoff and co-workers on singular integral equations and connections with Bäcklund transformations.

 

Integrable ordinary difference equations (OΔEs) were first extensively studied by Quispel, Roberts, and Thompson (QRT) in the second-order case, and by Papageorgiou, Nijhoff and Capel, and by Quispel et al. in the higher-order case.

 

As is not uncommon when one studies deformations of a theory, the discrete case exhibits both similarities to, as well as differences from, the continuous case.

 

Discrete integrable systems may be (in a sense) more fundamental than continuous ones. This is one reason for their significance. Not only does e.g. the continuous KdV equation arise in the continuum limit from the discrete KdV equation, but in fact, the entire infinite hierarchy of continuous KdV equations arises in the limit from the single discrete KdV equation! This was shown in an important, but little-known, paper by Wiersma and Capel using vertex operator methods. (Similarly for other integrable PΔEs.). Related to this is the fact that discrete integrable PΔEs exhibit a much higher symmetry than their continuous counterparts, this symmetry being broken in the continuum limit. While this symmetry of integrable PΔEs and of their Lax pairs under permutation of the (discrete) spatial independent variables has been very explicit since e.g. our early work, a much fuller understanding has been obtained more recently in terms of so-called multi-dimensional consistency.

 

In this talk, we will discuss autonomous and non-autonomous integrable PΔEs, and their reductions to autonomous and non-autonomous integrable OΔEs.

 

 

Geometry of quadrirational Yang-Baxter maps

Alexander Veselov

 

The quadrirational maps of P^1xP^1 into itself were introduced and studied by Adler, Bobenko and Suris. I will explain how one can use their results to describe when such maps satisfy the Yang-Baxter relation. The corresponding maps define the symmetries of degree 4 del Pezzo surfaces and have natural geometric interpretation in terms of pencils of quadrics.

 

The talk is based on a joint work with Papageorgiu, Tongas and Suris.

 

 

Recursion operators of the Viallet equation

Jing Ping Wang

 

In this talk, I'll present two recursion operators for the Viallet equation.They both can be written as the products of Hamiltonian and symplectic operators. They are related by the elliptic curve equation associated with the Viallet equation.

 

 

Symmetry algebra of discrete KdV equations and corresponding differential-difference equations of Volterra type

Pavlos Xenitidis

 

A sequence of canonical conservation laws for all the Adler-Bobenko-Suris equations is derived and is employed in the construction of a hierarchy of master symmetries for equations H1-H3, Q1-Q3. For the discrete potential and Schwarzian KdV equations it is shown that their local generalized symmetries and nonlocal master symmetries in each lattice direction form centerless Virasoro type algebras. In particular, for the discrete potential KdV, the structure of its symmetry algebra is explicitly given. Interpreting the hierarchies of symmetries of equations H1-H3, Q1-Q3 as differential-difference equations of Yamilov's discretization of Krichever-Novikov equation, corresponding hierarchies of master symmetries along with isospectral and non-isospectral zero curvature representations are derived for all of them.



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