A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (P∆Es) is reviewed. The method assumes that the P∆Es are defined on a quadrilateral, and consistent around the cube. Next, the method is extended to systems of P∆Es where one has to carefully account for equations defined on edges of the quadrilateral. Lax pairs are presented for scalar integrable P∆Es classified by Adler, Bobenko, and Suris and systems of P∆Es including the integrable 2-component potential Korteweg-de Vries lattice system, as well as nonlinear Schrödinger and Boussinesq-type lattice systems. Previously unknown Lax pairs are presented for P∆Es recently derived by Hietarinta (J. Phys. A: Math. Theor., 44 (2011) Art. No. 165204). The method is algorithmic and is being implemented in Mathematica.

We show, in full generality, that the staircase method [27,29] provides integrals for mappings, and correspondences, obtained as traveling wave reductions of (systems of) integrable partial difference equations. We apply the staircase method to a variety of equations, including the KortewegDe Vries equation, the five-point Bruschi-Calogero-Droghei equation, the QD-algorithm, and the Boussinesq system. We show that, in all these cases, if the staircase method provides r integrals for an n-dimensional mapping, with 2r < n, then one can introduce q ≤ 2r variables, which reduce the dimension of the mapping from n to q. These dimension-reducing variables are obtained as joint invariants of k-symmetries of the mappings. Our results support the idea that often the staircase method provides sufficiently many integrals for the periodic reductions of integrable lattice equations to be completely integrable. We also study reductions on other quad-graphs than the regular Z 2 lattice, and we prove linear growth of the multi-valuedness of iterates of high-dimensional correspondences obtained as reductions of the QD-algorithm.

We describe how to pose straight band initial value problems for lattice equations defined on arbitrary stencils. In finitely many directions, we arrive at discrete Goursat problems and in the remaining directions we find Cauchy problems. Next, we consider (s1, s2)-periodic initial value problems. In the Goursat directions, the periodic solutions are generated by correspondences. In the Cauchy directions, assuming the equation to be multi-linear, the periodic solution can be obtained uniquely by iteration of a particularly simple mapping, whose dimension is a piecewise linear function of s1, s2.

We construct and study certain Liouville integrable, superintegrable and non-commutative integrable systems, which are associated with multi-sums of products.

Abstract. The discrete-time Toda equation arises as a universal equation for the relevant Hankel determinants associated with one-variable orthogonal polynomials through the mechanism of adjacency, which amounts to the inclusion of shifted weight functions in the orthogonality condition. In this paper we extend this mechanism to a new class of two-variable orthogonal polynomials where the variables are related via an elliptic curve. This leads to a 'Higher order Analogue of the Discrete-time Toda' (HADT) equation for the associated Hankel determinants, together with its Lax pair, which is derived from the relevant recurrence relations for the orthogonal polynomials. In a similar way as the quotient-difference (QD) algorithm is related to the discrete-time Toda equation, a novel quotient-quotient-difference (QQD) scheme is presented for the HADT equation. We show that for both the HADT equation and the QQD scheme, there exists well-posed s-periodic initial value problems, for almost all s ∈ Z 2 . From the Lax-pairs we furthermore derive invariants for corresponding reductions to dynamical mappings for some explicit examples.

We give a method to calculate closed-form expressions in terms of multisums 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 all equations of the Adler-Bobenko-Suris classification, with the exception of Q 4 .

We derive an nth order difference equation as a dual of a very simple periodic equation, and construct ⌊(n + 1)/2⌋ explicit integrals and integrating factors of this equation in terms of multi-sums of products. We also present a generating function for the degrees of its iterates, exhibiting polynomial growth. In conclusion we demonstrate how the equation in question arises as a reduction of a system of lattice equations related to an integrable equation of Levi and Yamilov. These three facts combine to suggest the integrability of the nth order difference equation.

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 KdV (pKdV) equation. 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.