On Tautological Flows of Partial Difference Equations

We propose a new analyzing method, which is called the tautological flow method, to analyze the integrability of partial difference equations (P$\mathrm{\Delta }$Es) based on that of partial differential equations (PDEs). By using this method, we prove that the discrete $q$-KdV equation is a discrete symmetry of the $q$-deformed KdV hierarchy and its bihamiltonian structure, and we also demonstrate how to directly search for continuous symmetries and bihamiltonian structures of P$\mathrm{\Delta }$Es by using the approximated tautological flows and their quasi-triviality transformation.