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

journal contribution
posted on 2022-03-29, 03:26 authored by Pieter Van Der KampPieter Van Der Kamp, Gilles QuispelGilles Quispel
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 ndimensional mapping, with 2r < n, then one can introduce q ≤ 2r variables, which reduce the dimension of the mapping from n to q. These dimensionreducing 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 ℤ2 lattice, and we prove linear growth of the multi-valuedness of iterates of high-dimensional correspondences obtained as reductions of the QD-algorithm. © 2010 IOP Publishing Ltd.


This research has been funded by the Australian Research Council through the Centre of Excellence for Mathematics and Statistics of Complex Systems. We are grateful for the hospitality of the Isaac Newton Institute during the 2009 program on Discrete Integrable Systems.


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Journal of Physics A: Mathematical and Theoretical



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IOP Publishing



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© 2010 IOP Publishing Ltd.

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