Center for Scientific Workshops in All Disciplines

Current Workshop | Overview | Back | Home | Search | | ||||||||||

## Discrete Integrable Systems |

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.
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.
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.
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.
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.
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).
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.
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 Firstly, this is done by considering the discrete Schwarzian KP equation
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.
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.
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.
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.
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.
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.
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.
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 L
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.
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 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.
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.
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.
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. [Back] |