On a class of 2D integrable lattice equations

Abstract

We develop a new approach to the classification of integrable equations of the form $$u_{xy}=f(u,u_x,u_y,{△}_{z}u{△}_{\bar{z}}u,{△}_{z\bar{z}}u),$$ where ${△}_z$ and ${△}_{\bar{z}}$ are the forward/backward discrete derivatives. The following 2-step classification procedure is proposed: (1) First we require that the dispersionless limit of the equation is integrable, that is, its characteristic variety defines a conformal structure which is Einstein-Weyl on every solution. (2) Secondly, to the candidate equations selected at the previous step we apply the test of Darboux integrability of reductions obtained by imposing suitable cut-off conditions.

References in corpus (7)

• Einstein-Weyl geometry, the dKP equation and twistor theory
• A class of Einstein--Weyl spaces associated to an integrable system of hydrodynamic type
• On the Einstein-Weyl and conformal self-duality equations
• Integrable Background Geometries
• On a classification algorithm of the integrable two-dimensional lattices via Lie-Rinehart algebras
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Cited by in corpus (3)

• Characteristic Lie Algebras of Integrable Differential-Difference Equations in 3D
• Second-order PDEs in 3D with Einstein-Weyl conformal structure
• An algebraic criterion of the Darboux integrability of differential-difference equations and systems