On the singularity structure of the discrete KdV equation

TL;DR

This paper investigates the complex singularity structure of the discrete KdV equation through reductions and compares it with non-integrable cases, revealing subtle behaviors and refining understanding of its integrability properties.

Contribution

It provides a detailed analysis of the singularity patterns in the discrete KdV equation and demonstrates the applicability of the express method to higher order mappings.

Findings

Singularity structure is more subtle than simple confinement suggests.

The express method applies successfully to higher order mappings.

Existence of non-confining patterns does not imply non-integrability.

Abstract

The discrete KdV (dKdV) equation, the pinnacle of discrete integrability, is often thought to possess the singularity confinement property because it confines on an elementary quadrilateral. Here we investigate the singularity structure of the dKdV equation through reductions of the equation, obtained for initial conditions on a staircase with height 1, and show that it is much more subtle than one might assume. We first study the singularities for the mappings obtained after reduction and contrast these with the singularities that arise in non-integrable generalizations of those mappings. We then show that the so-called `express method' for obtaining dynamical degrees for second order mappings can be succesfully applied to all the higher order mappings we derived. Finally, we use the information obtained on the singularity structure of the reductions to describe an important subset of singularity patterns for the dKdV equation and we present an example of a non-confining pattern and explain why its existence does not contradict the integrability of the dKdV equation.