On the constrained discrete mKP hierarchies: gauge transformations and the generalized Wronskian solutions

We apply the gauge transformations $T_\mathrm{D}$ (differential type) and $T_\mathrm{I}$ (integral type) to study the discrete mKP hierarchies. We prove that $T_\mathrm{D}$ and $T_\mathrm{I}$ can be commutative and the product of $T_\mathrm{D}$ and $T_\mathrm{I}$ satisfies the Sato equation. By means of gauge transformations, we arrive at the necessary and sufficient condition for reducing the generalized Wronskian solutions to constrained hierarchies. Finally, we give an example in the Appendix.