Abstract:

We give a method to calculate closed-form expressions in terms of multi-sums of products for integrals of ordinary difference equations which are obtained as traveling wave reductions of integrable partial difference equations. Important ingredients are the staircase method, a non-commutative Vieta formula, and certain splittings of the Lax matrices. The method is applied to the Lotka-Volterra, Korteweg-De Vries (KdV), modified KdV, sine-Gordon equations and all equations of the Adler-Bobenko-Suris classification, with the exception of $Q_{4}.$