Modified constrained KP hierarchy and bi-Hamiltonian structures

Abstract

We consider new modifications for the constrained KP hierarchy associated with a sequence of energy-dependent Lax operators. When $$n=1, 2,$$ n = 1 , 2 , the hierarchies are shown to be the KN hierarchy and a modified Yajima–Oikawa hierarchy respectively. When $$n=3$$ n = 3 , it is the hierarchy of a new modified Melnikov type with a bi-Hamiltonian structure.