Integrable M\"obius invariant evolutionary lattices of second order

TL;DR

This paper classifies second-order integrable lattices invariant under Möbius transformations, identifying five equations including three new ones, and establishes connections with known lattices through Miura-type substitutions.

Contribution

It provides a complete classification of Möbius-invariant integrable lattices of second order, introducing three new equations and relating them to existing models.

Findings

Identified five Möbius-invariant integrable lattices, including three new equations.

Established difference Miura-type substitutions linking these lattices to known polynomial models.

Presented classification results for generic lattices.

Abstract

We solve the classification problem for integrable lattices of the form $u_{,t}=f(u_{-2},\dots,u_2)$ under the additional assumption of invariance with respect to the group of linear-fractional transformations. The obtained list contains 5 equations, including 3 new. Difference Miura type substitutions are found which relate these equations with known polynomial lattices. We also present some classification results for the generic lattices.