Lax Pair for a Novel Two-Dimensional Lattice

Abstract

In paper by I.T. Habibullin and our joint paper the algorithm for classification of integrable equations with three independent variables was proposed. This method is based on the requirement of the existence of an infinite set of Darboux integrable reductions and on the notion of the characteristic Lie-Rinehart algebras. The method was applied for the classification of integrable cases of different subclasses of equations u n,xy = f (u n+1 , u n , u n−1 , u n,x , u n,y ) of special forms. Under this approach the novel integrable chain was obtained. In present paper we construct Lax pair for the novel chain. To construct the Lax pair, we use the scheme suggested in papers by E.V. Ferapontov. We also study the periodic reduction of the chain.

Integrability conditions for two-dimensional Toda-like equations

“…5 On reductions of the lattice (E7) Lattice (E7) was found in [13] as a result of the integrable classification of nonlinear lattices. In [20] a Lax pair was found for this equation…”

“…Equation (E7) was obtained during the classification process in [13]. The Lax pair for it was found in [20]. For the lattices (E6), (E7) the reductions of the form (1.4) are investigated in [21] and correspondingly in [13] admitting finite-dimensional characteristic Lie algebras for arbitrary m. However in these articles the integrals are not presented.…”