Integrability of reductions of the discrete KdV and potential KdV equations

TL;DR

This paper investigates the integrability of certain mappings derived from the discrete KdV and potential KdV equations, demonstrating their complete integrability in the Liouville-Arnold sense.

Contribution

It establishes the complete integrability of mappings from the discrete KdV and two copies of the pKdV equations, expanding understanding of their integrable structures.

Findings

Mappings from discrete KdV are completely integrable.

Mappings from two copies of pKdV are integrable.

Integrability is shown in the Liouville-Arnold sense.

Abstract

We study the integrability of mappings obtained as reductions of the discrete Korteweg-de Vries (KdV) equation and of two copies of the discrete potential Korteweg-de Vries equation (pKdV). We show that the mappings corresponding to the discrete KdV equation, which can be derived from the latter, are completely integrable in the Liouville-Arnold sense. The mappings associated with two copies of the pKdV equation are also shown to be integrable.