The generalized Wronskian solutions of the constrained mKP hierarchy

In this paper, we investigate the $(k, m)$-constrained 1st modified Kadomtsev-Petviashvili (mKP) hierarchy $(L^k)_{\leq 0}= \sum_{i=1}^m q_i \partial^{-1} r_i \partial$. Here, we obtain the corresponding solutions in the form of generalized Wronskians, which include the Wronskians and Grammians as special cases. Most importantly, these generalized Wronskian solutions are proved to satisfy the bilinear equations of the $(k, m)$-constrained mKP hierarchy, which is generally nontrivial. Our results here will be helpful in the derivation of the more general addition formulae and polynomial solutions for the 1st mKP hierarchy.