Hamiltonian structures of nonlinear evolution equations connected with a polynomial pencil

Abstract

For a generalized Zakharov-Shabat system in which the matrix potential is a polynomial in the spectral parameter a generating operator is constructed which makes it possible to compactly write out the nonlinear evolution equations (NEE) connected with the system. The eigenfunctions of the generating operator — the “squares“ of solutions of the original system — are found. The Hamiltonian property of the NEE and the existence of a hierarchy of Hamiltonian structures are established.