Lax integrable nonlinear partial difference equations

•Lax pair formalism provides an effective tool to identify integrable nonlinear partial difference equations.•A new nonlinear partial difference equation has been reported, which does not possess the well known CAC property.•A nonlinear partial difference equation that can be transformed into a linear partial difference equation has been reported.•A nonlinear partial difference equation which is a ΔΔΔΔK-dV equation has been reported.•Several higher dimensional nonlinear difference equations or mappings are reported.

A systematic investigation to derive nonlinear lattice equations governed by partial difference equations admitting specific Lax representation is presented. Further whether or not the identified lattice equations possess other characteristics of integrability namely Consistency Around the Cube (CAC) property and linearizability through a global transformation is analyzed. Also it is presented that how to derive higher order ordinary difference equations or mappings from the obtained lattice equations through periodic reduction and investigated whether they are measure preserving or linearizable and admit sufficient number of integrals leading to their integrability.