Gradient theorem for completely integrable Hamiltonian systems

Abstract

For evolution equations which can be written in Hamiltonian form two ways, there exists a relation between two functions Q(1) and Q(2), both of which are gradients of conserved functionals. The relation can be extended to define (recursively) functions Q(n). It is shown that the Q(n) corresponding to the general evolution equation associated with the Zakharov–Shabat eigenvalue problem are all gradients of conserved functionals. This in turn implies all these functionals are in involution.