Abstract
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.
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