Involutivity of integrals for sine-Gordon, modified KdV and potential KdV maps

TL;DR

This paper proves the involutivity of integrals for sine-Gordon, modified KdV, and potential KdV maps, derived from partial difference equations, using explicit Poisson bracket calculations within their symplectic structures.

Contribution

It establishes the involutivity of integrals for these maps, providing explicit Poisson bracket formulas and confirming their symplectic properties.

Findings

Integrals are involutive with respect to symplectic structures.

Explicit Poisson brackets between multi-sums are derived.

Supports the integrability of the studied maps.

Abstract

Closed form expressions in terms of multi-sums of products have been given in \cite{Tranclosedform, KRQ} of integrals of sine-Gordon, modified Korteweg-de Vries and potential Korteweg-de Vries maps obtained as so-called $(p,-1)$-traveling wave reductions of the corresponding partial difference equations. We prove the involutivity of these integrals with respect to recently found symplectic structures for those maps. The proof is based on explicit formulae for the Poisson brackets between multi-sums of products.