Full-deautonomisation of a lattice equation

TL;DR

This paper extends the full-deautonomisation method to discrete lattice equations and their reductions, enabling prediction of algebraic entropy and integrability properties for complex non-linear systems.

Contribution

It demonstrates the novel application of full-deautonomisation to lattice equations and higher-order mappings, predicting algebraic entropy and nonintegrability.

Findings

Successfully applied to a nonintegrable lattice equation with confined singularities.

Predicted algebraic entropy for higher-order nonconfining mappings.

Extended the method's applicability beyond second-order mappings.

Abstract

In this letter we report on the unexpected possibility of applying the full-deautonomisation approach we recently proposed for predicting the algebraic entropy of second-order birational mappings, to discrete lattice equations. Moreover, we show, on two examples, that the full-deautonomisation technique can in fact also be successfully applied to reductions of these lattice equations to mappings with orders higher than 2. In particular, we apply this technique to a recently discovered lattice equation that has confined singularities while being nonintegrable, and we show that our approach accurately predicts this nonintegrable character. Finally, we demonstrate how our method can even be used to predict the algebraic entropy for some nonconfining higher order mappings.