Features of Discrete Integrability
Claude M. Viallet
Abstract We describe some standard features of integrability for a class of
integrable partial difference equations: the quad equations. These features are
the existence of Lax pairs, higher dimensional consistency, singularity properties,
existence of symmetries, and low complexity (vanishing algebraic entropy). All
these features have pros and cons, and we give a glimpse of them.
Keywords Discrete integrability
1 Introduction
For ages we have been inclined to think of evolution equations as differential
equations, the discrete versions coming at a later stage, in particular when one is
constructing a numerical scheme for their resolution. What gained in the recent
years is the consideration of discrete equations per se. There are a number of reasons
for this change. One is the considerable increase of the computational power of our
machines, especially for the formal calculus. Another one is the advent of discrete
equations in diverse branches of theoretical physics and mathematics, from 2D
gravity [1–3] to statistical models on the lattice [4–7], not forgetting the fundamental
contribution of [8], which gave its letters of nobility to the study of self-maps of
spaces of finite dimension.
A special interest was taken in the integrable cases, their rich structure leading to
new developments. One basic question arising immediately: given a discrete system,
in the form of a recurrence relation, a discrete time evolution, or a lattice equation,
which are the discrete forms of ordinary and partial differential equations, how do
we characterise its integrability?
C. M. Viallet ()
Centre National de la Recherche Scientifique, Sorbonne Université, Paris, France
e-mail: claude.viallet@upmc.fr
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_2
21
Claude M. Viallet
Abstract We describe some standard features of integrability for a class of
integrable partial difference equations: the quad equations. These features are
the existence of Lax pairs, higher dimensional consistency, singularity properties,
existence of symmetries, and low complexity (vanishing algebraic entropy). All
these features have pros and cons, and we give a glimpse of them.
Keywords Discrete integrability
1 Introduction
For ages we have been inclined to think of evolution equations as differential
equations, the discrete versions coming at a later stage, in particular when one is
constructing a numerical scheme for their resolution. What gained in the recent
years is the consideration of discrete equations per se. There are a number of reasons
for this change. One is the considerable increase of the computational power of our
machines, especially for the formal calculus. Another one is the advent of discrete
equations in diverse branches of theoretical physics and mathematics, from 2D
gravity [1–3] to statistical models on the lattice [4–7], not forgetting the fundamental
contribution of [8], which gave its letters of nobility to the study of self-maps of
spaces of finite dimension.
A special interest was taken in the integrable cases, their rich structure leading to
new developments. One basic question arising immediately: given a discrete system,
in the form of a recurrence relation, a discrete time evolution, or a lattice equation,
which are the discrete forms of ordinary and partial differential equations, how do
we characterise its integrability?
C. M. Viallet ()
Centre National de la Recherche Scientifique, Sorbonne Université, Paris, France
e-mail: claude.viallet@upmc.fr
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_2
21