Abstract
We develop a new approach to the classification of integrable equations of the form uxy=f(u,ux,uy,△zu△z¯u,△zz¯u), where △z and △z¯ are the forward/backward discrete derivatives. The following two-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) Second, to the candidate equations selected at the previous step, we apply the test of Darboux integrability of reductions obtained by imposing suitable cutoff conditions.
Highlights
In this paper we develop a new approach to the classification of integrable lattice type equations in 3D by combining the geometric approach of [9] with the test of [10, 11] based on the requirement of Darboux integrability of suitably reduced equations
To these candidate equations we apply the test of Darboux integrability of reductions obtained by imposing suitable cut-off conditions as proposed in [10, 11]
Our main result is a complete list of integrable equations of type (1)
Summary
In this paper we develop a new approach to the classification of integrable lattice type equations in 3D by combining the geometric approach of [9] with the test of [10, 11] based on the requirement of Darboux integrability of suitably reduced equations. Note that dispersionless limits of the above equations (obtained as → 0) coincide with the Boyer-Finley equation uxy = euzz and the equation uxy ux uy u2z uzz Both limits belong to the class of dispersionless integrable PDEs. Further integrable examples of type (1) obtained in [10] include the equations uxy = (ux − u)(uy − u) zzu zu zu. (2) Secondly, replacing uz and uzz in the equations obtained at the previous step by zu zu and zzu, respectively, we obtain equations of type (1) which, at this stage, are our candidates for integrability. To these candidate equations we apply the test of Darboux integrability of reductions obtained by imposing suitable cut-off conditions as proposed in [10, 11].
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