The constrained KP hierarchy and the bigraded Toda hierarchy of $(M,1)$-type

Abstract: In this paper, we extend the matrix-resolvent method to the study of the Dubrovin--Zhang type tau-functions for the constrained KP hierarchy and the bigraded Toda hierarchy of $(M,1)$-type. We show that the Dubrovin--Zhang type tau-function of an arbitrary solution to the bigraded Toda hierarchy of $(M,1)$-type is a Dubrovin--Zhang type tau-function for the constrained KP hierarchy, which generalizes the result in [10, 35] for the Toda lattice hierarchy and the NLS hierarchy corresponding to the $M=1$ case.