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

TL;DR

The paper demonstrates that the staircase method systematically provides integrals for periodic reductions of integrable lattice equations, enabling the reduction of high-dimensional mappings and supporting their complete integrability.

Contribution

It generalizes the staircase method for integrals, applies it to various equations, and introduces a dimension reduction technique via joint invariants of symmetries.

Findings

The staircase method yields integrals for multiple integrable equations.

Dimension reduction from n to q variables is possible using joint invariants.

Reductions on non-regular quad-graphs exhibit linear growth in multi-valuedness.

Abstract

We show, in full generality, that the staircase method 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 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 2D lattice, and we prove linear growth of the multi-valuedness of iterates of high-dimensional correspondences obtained as reductions of the QD-algorithm.