Algebraic entropy computations for lattice equations: why initial value problems do matter

TL;DR

This paper demonstrates that the algebraic entropy calculations for lattice equations are highly sensitive to the choice of initial value problems, challenging the common belief that integrable equations always exhibit polynomial growth.

Contribution

It reveals the dependence of degree growth on initial value configurations and proposes a new approach for integrability testing based on single initial value growth.

Findings

Hirota's discrete KdV equation shows exponential growth for certain initial conditions.

Discrete Liouville equation exhibits different growth behaviors depending on initial conditions.

Proposes using single initial value growth for more reliable integrability tests.

Abstract

In this letter we show that the results of degree growth (algebraic entropy) calculations for lattice equations strongly depend on the initial value problem that one chooses. We consider two problematic types of initial value configurations, one with problems in the past light-cone, the other one causing interference in the future light-cone, and apply them to Hirota's discrete KdV equation and to the discrete Liouville equation. Both of these initial value problems lead to exponential degree growth for Hirota's dKdV, the quintessential integrable lattice equation. For the discrete Liouville equation, though it is linearizable, one of the initial value problems yields exponential degree growth whereas the other is shown to yield non-polynomial (though still sub-exponential) growth. These results are in contrast to the common belief that discrete integrable equations must have polynomial growth and that linearizable equations necessarily have linear degree growth, regardless of the initial value problem one imposes. Finally, as a possible remedy for one of the observed anomalies, we also propose basing integrability tests that use growth criteria on the degree growth of a single initial value instead of all the initial values.