Features of Discrete Integrability
33
5 Comments and Perspectives
We have given a glimpse of the subject, leaving aside some of the recent developments like, for example, the Lagrangian multiform approach [9, 45, 46], which is a
promising approach.
Our point was rather to give an idea of some of the features of integrability, with
their strength and their limitations, which we can summarise as follows:
– Lax pairs. pro: Lax pairs are a powerful instrument to produce solutions, as they
were in the continuous case; con: They are not always easy to find, and they may
have different forms (the size of the linear system is not known in advance, and
there may be different pairs of different sizes). In addition one should beware of
fakes [47, 48].
– 3D consistency. pro: provides a Lax pair, and allows a classification; con: not
equivalent to integrability.
– Singularity confinement. pro: constructive and allowing classification at least for
the order two equations; con: not necessary nor sufficient for integrability.
– Symmetries. pro: constructive and necessary; con: not always sufficient as one
needs a sufficiently large number of them.
– Algebraic entropy. pro: it is canonical (invariant by birational changes of
coordinates), and the vanishing of the entropy may serve as a characterisation of
integrability, as the sign of catastrophic drop of the complexity, con: destructive
rather than constructive, since it gives a yes/no answer to the question “is this
model integrable?”
We have alluded to the relation of the singularity structure with the measure of
complexity of the evolutions. What is still missing is a better understanding of its
relations with Lax pairs and Symmetries.
One hint may come from the following fact: when analysing the singularities, the
sequence of degrees of the iterates, as well as the symmetries, the most important
relations bear on local properties with a finite extension. This means that, even in
the discrete world, we may distinguish three scales—as we have in the continuous
world—infinitesimal, local, global. Infinitesimal would be the defining relation (one
cell), local would be any relation extending over more than one cell, and of course
global would be extending to infinity. What we see is that we can reach a conclusion
on a global property like integrability from local ones. The algebraic nature of
the models we consider is probably at the origin of this phenomenon. Since this
algebraic nature also conditions the singularity structure, we should look further
into relations of the singularities with symmetries and even Lax pairs.
We have work to do.
Acknowledgments The author would like to thank the organisers of the 11th symposium “Théorie
Quantique et Symétries” for the invitation at the Centre de Recherches Mathématiques, in
particular at the occasion of the “Special Session in honour of Decio Levi: Integrability: continuous
and discrete, classical & quantum.”
33
5 Comments and Perspectives
We have given a glimpse of the subject, leaving aside some of the recent developments like, for example, the Lagrangian multiform approach [9, 45, 46], which is a
promising approach.
Our point was rather to give an idea of some of the features of integrability, with
their strength and their limitations, which we can summarise as follows:
– Lax pairs. pro: Lax pairs are a powerful instrument to produce solutions, as they
were in the continuous case; con: They are not always easy to find, and they may
have different forms (the size of the linear system is not known in advance, and
there may be different pairs of different sizes). In addition one should beware of
fakes [47, 48].
– 3D consistency. pro: provides a Lax pair, and allows a classification; con: not
equivalent to integrability.
– Singularity confinement. pro: constructive and allowing classification at least for
the order two equations; con: not necessary nor sufficient for integrability.
– Symmetries. pro: constructive and necessary; con: not always sufficient as one
needs a sufficiently large number of them.
– Algebraic entropy. pro: it is canonical (invariant by birational changes of
coordinates), and the vanishing of the entropy may serve as a characterisation of
integrability, as the sign of catastrophic drop of the complexity, con: destructive
rather than constructive, since it gives a yes/no answer to the question “is this
model integrable?”
We have alluded to the relation of the singularity structure with the measure of
complexity of the evolutions. What is still missing is a better understanding of its
relations with Lax pairs and Symmetries.
One hint may come from the following fact: when analysing the singularities, the
sequence of degrees of the iterates, as well as the symmetries, the most important
relations bear on local properties with a finite extension. This means that, even in
the discrete world, we may distinguish three scales—as we have in the continuous
world—infinitesimal, local, global. Infinitesimal would be the defining relation (one
cell), local would be any relation extending over more than one cell, and of course
global would be extending to infinity. What we see is that we can reach a conclusion
on a global property like integrability from local ones. The algebraic nature of
the models we consider is probably at the origin of this phenomenon. Since this
algebraic nature also conditions the singularity structure, we should look further
into relations of the singularities with symmetries and even Lax pairs.
We have work to do.
Acknowledgments The author would like to thank the organisers of the 11th symposium “Théorie
Quantique et Symétries” for the invitation at the Centre de Recherches Mathématiques, in
particular at the occasion of the “Special Session in honour of Decio Levi: Integrability: continuous
and discrete, classical & quantum.”