Darboux transformations for linear operators on two dimensional regular lattices

Darboux transformations for linear operators on regular two dimensional lattices are reviewed. The six point scheme is considered as the master linear problem, whose various specifications, reductions, and their sublattice combinations lead to other linear operators together with the corresponding Darboux transformations. The second part of the review deals with multidimensional aspects of (basic reductions of) the four point scheme, as well as the three point scheme.

💡 Research Summary

This review paper provides a comprehensive synthesis of Darboux transformations (DTs) for linear operators defined on regular two‑dimensional lattices, positioning the six‑point scheme as the master linear problem from which virtually all other discrete linear equations of interest can be derived. The authors first introduce the most general second‑order difference equation involving a lattice point ((m,n)) and its six nearest neighbours. By allowing the six coefficient functions (a_{m,n},\dots,f_{m,n}) to be arbitrary, the six‑point scheme simultaneously encompasses discrete analogues of the Laplace, Moutard, and Kadomtsev–Petviashvili (KP) equations.

The core of the Darboux construction is presented in a unified “ratio‑type” form. Given a particular solution (\psi) of the six‑point equation and an auxiliary solution (\phi) (the seed), a new solution (\tilde\psi) is generated by
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