Duality in Discrete Integrable Systems
J. M. Tuwankotta
Department of Mathematics, Bandung Institute of Technology
Discrete dynamical systems arises quite naturally in applications, for
example being an approximation for Systems of Ordinary of Partial
Differential equations. The theory however, is far less well understood in
comparison with the continuous cases. Integrability, on the other hand is
also a well defined and rather fully understood concept in Systems of
Ordinary Differential Equations (but not so much in Partial Differential
Equations). In the discrete cases, the situation is more or less
One might argue that Integrable Systems are not so many in compare with the
nonintegrable systems, or even less exciting due to the discoverey and the
development of the Theory of Chaos in 1960s. However, integrable systems is
mathematically interesting due its richness in structures.
Moreover, it serves as a very good approximation for the more exciting
nonintegrable systems and most of the time, it can give insight to
underlaying complex mechanism that is found in the nonitegrable systems.
The year of 1988 and 1989, two papers are published describing a two
dimensonal rational maps depends on 18 parameters, the celebrated
Quispel-Roberts-Thompson (QRT) map (from La Trobe University, University New
South Wales, and Australian National University respectively). This turns
out to be one of the most general integrable system in two dimension
(arguably the most general). In 2010, the late J.J. Duistermaat (Utrecht
Univesity) published a seminal book with the title Discrete Integrable
Systems: QRT Maps and Elliptic Surfaces, where various different angle and
approaches have been considered in understanding the QRT maps.
In this talk, we will try to uncovered a relatively new concept in Discrete
Integrable Systems which is Duality. This concept is introduced by G.R.W.
Quispel (La Trobe Univesity) in 2005, for a discrete dynamical system
defined by a single equation. This concept is easily generalized to
discrete dynamical systems defined by a system of equations, however the
procedure of getting the duals is far from trivial extension of the known
one. We generalize this method, and propose a different, rather more
straight forward, approach for computing the duals.