The Gauge Transformations And Bilinear Equations Of The Constrained MKP Hierarchy

In this paper,we take the constrained mKP hierarchy(?)with the Kupershmidt-Kiso version as the research object,and consider the basic prop-erties of the k-constrained mKP hierarchy with m components:Wronskian reduc-tion,gauge transformations,Hirota bilinear equations,Sato-Backlund transformations,Boson-Fermion correspondence and other related issues.The main results are as fol-lows:(1)The necessary and sufficient conditions for the constrained mKP hierarchy from the Wronskian solutions of the mKP hierarchy are given,and the corresponding example is given.(2)Two types of gauge transformations for the constrained mKP hierarchy are s-tudied.The corresponding gauge transformations are required to keep not only the Lax equation but also the Lax operator.For this,by selecting the special generating eigen-function and adjoint eigenfunction,the elementary gauge transformation operators of the mKP hierarchy(?)become the ones in the constrained case.And the corresponding successive applications of TD and TI on the eigenfunction ? and the adjoint eigenfunction ? are discussed.(3)The bilinear equations are equivalent forms of the constrained mKP hierarchy.By introducing two auxiliary functions ? and ?,the corresponding equations are written in Hirota forms.The specific solution forms of the auxiliary functions ? and ? are given by using gauge transformations.Furthermore,based on bilinear equations,Sato-Backlund transformations of mKP hierarchy and its constrained cases are constructed.(4)The Hirota bilinear equations of the constrained mKP hierarchy are expressed as the vacuum expectation values of Clifford operators by using the free fermion method of the mKP hierarchy.Then we mainly use the Boson-Fermion correspondence to solve the Hirota bilinear equations of the k-constrained mKP hierarchy.Further,by choosing special group elements in G,the corresponding rational solutions and soliton solutions are given.