Peter H. van der Kamp

The staircase method: integrals for periodic reductions of integrable lattice equations

We show, in full generality, that the staircase method (Papageorgiou et al 1990 Phys. Lett. A 147 106?14, Quispel et al 1991 Physica A 173 243?66) 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 Korteweg?De Vries equation, the five-point Bruschi?Calogero?Droghei equation, the quotient-difference (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 , 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 lattice, and we prove linear growth of the multi-valuedness of iterates of high-dimensional correspondences obtained as reductions of the QD-algorithm.

Initial value problems for lattice equations

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.

Symbolic Computation of Lax Pairs of Partial Difference Equations using Consistency Around the Cube

A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (PΔEs) is reviewed. The method assumes that the PΔEs are defined on a quadrilateral, and consistent around the cube. Next, the method is extended to systems of PΔEs where one has to carefully account for equations defined on edges of the quadrilateral. Lax pairs are presented for scalar integrable PΔEs classified by Adler, Bobenko, and Suris and systems of PΔEs including the integrable two-component potential Korteweg–de Vries lattice system, as well as nonlinear Schrödinger and Boussinesq-type lattice systems. Previously unknown Lax pairs are presented for PΔEs recently derived by Hietarinta (J. Phys. A, Math. Theor. 44:165204, 2011). The method is algorithmic and is being implemented in Mathematica.

Closed-form expressions for integrals of traveling wave reductions of integrable lattice equations

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 all equations of the Adler–Bobenko–Suris classification, with the exception of Q4.

Twisted reductions of integrable lattice equations, and their Lax representations

It is well known that from two-dimensional lattice equations one can derive one-dimensional lattice equations by imposing periodicity in some direction. In this paper we generalize the periodicity condition by adding a symmetry transformation and apply this idea to autonomous and non-autonomous lattice equations. As results of this approach, we obtain new reductions of the discrete potential Korteweg–de Vries (KdV) equation, discrete modified KdV equation and the discrete Schwarzian KdV equation. We will also describe a direct method for obtaining Lax representations for the reduced equations.

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.

Sufficient number of integrals for the pth-order Lyness equation

In this communication, we present a sufficient number of first integrals for the Lyness equation of arbitrary order. We first use the staircase method (Quispel et al 1991 Physica A 173 243–66) to construct integrals of a derivative equation of the Lyness equation. Closed-form expressions for the integrals are given based on a non-commutative Vieta expansion. The integrals of the Lyness equation then follow directly from these integrals. Previously found integrals for the Lyness equation arise as special cases of our new set of integrals.

Growth of degrees of integrable mappings

We study mappings obtained as s-periodic reductions of the lattice Korteweg–de Vries equation. For small , we establish upper bounds on the growth of the degree of the numerator of their iterates. These upper bounds appear to be exact. Moreover, we conjecture that for any s 1, s 2 that are co-prime, the growth is , except when , where the growth is linear . Also, we conjecture the degree of the nth iterate in projective space to be .We study mappings obtained as s-periodic reductions of the lattice Korteweg–de Vries equation. For small , we establish upper bounds on the growth of the degree of the numerator of their iterates. These upper bounds appear to be exact. Moreover, we conjecture that for any s 1, s 2 that are co-prime, the growth is , except when , where the growth is linear . Also, we conjecture the degree of the nth iterate in projective space to be .

Integrable and superintegrable systems associated with multi-sums of products.

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

From discrete integrable equations to Laurent recurrences

We show how to obtain relations for the divisors of terms generated by a homogenized version of a rational recurrence. When the rational recurrence confines singularities the relations take the form of a rational recurrence, possibly with periodic coefficients. As the recurrence generates polynomials one expects it to possess the Laurent property. The method we develop uses ultra-discretization and recursive factorization. It is applied to certain QRT-maps which gives rise to Somos-k () sequences with periodic coefficients. Novel -rd order recurrences are obtained from the Nth order DTKQ-equation (). In each case the resulting recurrence equation has the Laurent property. The method is equally applicable to non-integrable or non-confining equations. However, in the latter case the degree and the order of the relation might display unbounded growth. We demonstrate the difference, by considering different parameter choices in a generalized Lyness equation.