Discrete Crum's theorems and lattice KdV-type equations

We develop Darboux transformations ($DTs$) and their associated Crum's formulas for two Schrödinger-type difference equations that are themselves discretized versions of the spectral problems of the KdV and modified KdV equations. With DTs viewed as a discretization process, classes of semidiscrete and fully discrete KdV-type systems, including the lattice versions of the potential KdV, potential modified KdV, and Schwarzian KdV equations, arise as the consistency condition for the differential/difference spectral problems and their DTs. The integrability of the underlying lattice models, such as Lax pairs, multidimensional consistency, $\tau$-functions, and soliton solutions, can be easily obtained by directly applying the discrete Crum's formulas.