MODIFYING THE KP, THE N(TH) CONSTRAINED KP HIERARCHIES AND THEIR HAMILTONIAN STRUCTURES

The Kadomtsev-Petviashvili (KP) hierarchy has infinitely many Hamiltonian pairs, the n(th) pair of them is associated with L(n), where L is the pseudodifferential operator (PDO) [3, 4]. In this paper, by the factorization L(n) = L(n) ... L(1) with L(j), j = 1,...,n being the independent PDOs, we construct the Miura transformation for the KP, which leads to a decomposition of the second Hamiltonian structure in the n(th) pair to a direct sum. Each term in the sum is the second structure in the initial pair associated with L(j). When we impose a constraint (1.9) (i.e. a new type of reduction) to the KP hierarchy, we obtain the similar results for the constrained KP hierarchy. In particular the second Hamiltonian structure for this hierarchy is transformed to a vastly simpler one.