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

24 pages

References in corpus (2)

• Classification of integrable equations on quad-graphs. The consistency approach
• Three Dimensional Integrable Mappings

Cited by in corpus (4)

• Liouville integrability and superintegrability of a generalized Lotka-Volterra system and its Kahan discretization
• Integrable and superintegrable systems associated with multi-sums of products
• Integrability of reductions of the discrete KdV and potential KdV equations
• Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation