On additional motion invariants of classical Hamiltonian wave systems (Q1106410)

It is shown that the existence of an analytic invariant in addition to the natural ones (moment, energy and others) for a classical Hamiltonian wave system leads to the existence of infinitely many such invariants. However this phenomenon reduces a Hamiltonian system to a complete integrable system only if the dispersion law is non-degenerative. Results on degenerative dispersion laws and a factorizing of the S-matrix are presented. Concrete examples of Hamiltonian systems are adduced, for instance \(i\psi_ t-\psi_{xx}+\psi_{yy}=| \psi |^ 2\psi.\) Also the singular elements of the scattering matrix are considered.