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

Closed form expressions in terms of multi-sums of products have been given in Tran et al (2009 J. Phys A: Math. Theor. 42 225201) and van der Kamp et al (2007 J. Phys. A: Math. Theor. 39 12789-98) of integrals of sine-Gordon, modified Korteweg-de Vries and potential Korteweg-de Vries maps obtained as so-called (p, -1)-travelling 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. © 2011 IOP Publishing Ltd.