Asymmetric integrable quad-graph equations

Abstract

Integrable difference equations commonly have more low-order conservation laws than occur for nonintegrable difference equations of similar complexity. We use this empirical observation to sift a large class of difference equations, in order to find candidates for integrability. It turns out that all such candidates have an equivalent affine form. These are tested by calculating their algebraic entropy. In this way, we have found several types of integrable equations, one of which seems to be entirely unrelated to any known discrete integrable system. We also list all single-tile conservation laws for the integrable equations in the above class.

Keywords:

• difference equations
• integrable systems
• conservation laws
• algebraic entropy

AMS Subject Classifications:

• 37K10
• 39A12
• 37K05

Acknowledgements

This work was carried out while we were visiting the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK. We thank the organisers of the programme Discrete Integrable Systems (January–June 2009) for the opportunity to participate. We particularly thank Frank Nijhoff, Kenichi Maruno, Reinout Quispel and James Atkinson for their helpful comments and Chris Hydon for assistance with Figure 1. We are also grateful to an eagle-eyed referee.

Notes

1. If but ≠ 0, similar calculations lead to precisely the same classifying equation (14), so nothing is lost by this assumption.

2. We thank Frank Nijhoff and Kenichi Maruno for alerting us to this.

3. We are grateful to an anonymous referee for this observation.