On the Classification of Discrete Hirota-Type Equations in 3D

Abstract

In the series of recent publications [14,15,17,20] we have proposed a novel approach to the classification of integrable differential/difference equations in 3D based on the requirement that hydrodynamic reductions of the corresponding dispersionless limits are 'inherited' by the dispersive equations. In this paper we extend this to the fully discrete case. Based on the method of deformations of hydrodynamic reductions, we classify discrete 3D integrable Hirota-type equations within various particularly interesting subclasses. Our method can be viewed as an alternative to the conventional multi-dimensional consistency approach.MSC: 35Q51, 37K10.

On the Einstein-Weyl and conformal self-duality equations

“…for which the EW equations reduce to the scalar second-order PDE u xx + u yy − (ln(e u − 1)) yy − (ln(e u − 1)) tt = 0. This is the dispersionless limit of the 'gauge-invariant' Hirota equation [21].…”

“…A Lax pair for this equation has recently been found in [27]. Its 2nd heavenly form analogous to (21) has been given in [12].…”

Direct linearisation of the discrete-time two-dimensional Toda lattices

“…which appeared very recently in [4] (after a recombination of all the discrete shifts). The parametrisation in (3.15a), however, describes its solution structure in a more natural way and allows to consider its continuum limit.…”

Explicit solutions for a nonlinear model on the honeycomb and triangular lattices

“…We consider our equations as a three-dimensional problem, bearing in mind that at some stage we have to take into account the fact that ∑ 3 i=1 e e e i = 0 0 0. The 'three-dimensionality' of the equations that we want to solve does not provide insuperable problems: there are some already known integrable systems in dimensions higher than two (see, for example, [5,16] for the lists of integrable discrete three-dimensional equations) and in what follows we use one of them. The second moment, stemming from the necessity to ensure the triviality of the superposition of the translations corresponding to ∑ 3 i=1 e e e i , turns out to be more embarrassing: a straightforward application of corresponding restrictions can drastically narrow the family of available solutions (we return to this question in the next sections when discussing the specific solutions).…”

“…τt ′ α,β ,γ + ρs ′′ α,β ,γ + a α,β T γ τ t α,β − a α,γ T β τ t α,γ + a β ,γ (T α τ) t β ,γ = (T α τ) T β τ T γ τ o α,β ,γ (A. 16) where o α,β ,γ is the right-hand side of equation (4.10).…”