Symmetry constraints of the KP hierarchy

Abstract

The authors study the KP hierarchy u_t= delta _xS_n+1 and the associated linear problems psi _t=B_n psi together with their adjoints psi _t*=-B_n* psi . Their hypothesis is that the triples u_t= delta _xS_n+1, psi _t=B_n psi , psi _t*=-B_n* psi * form a hierarchy of commuting equations with conserved densities S_n if they impose the constraint psi psi *=S_m. The general claim seems not so easy to prove. The authors confine themselves to consideration of a few cases. As a result they obtain a new non-trivial symmetry and a conserved density of the Mel'nikov system which happens to be a generalized Drinfeld-Sokolov System.