Lax pairs and Hamiltonians for the Kaup-Kupershmidt-type equation

This paper considers two types of nonlinear partial differential equations, namely the Kaup-Kupershmidt-type (KK-type) equations (KK-I and KK-II), which are of fifth order in space. We propose a Korteweg-de Vries (KdV)-type hierarchy relation and the associated differential operators to show that the KK-type equation can be regarded as a generalization of the KdV-type equation. We also demonstrate the canonical structures for both equations such as the Hamiltonian system, Poisson structure as well as the Ablowitz-Kaup-Newell-Segur structure. The Lax formulations for the KK-type equations are also presented. (paper)