Square eigenfunction approach to the hamiltonian structure for a new hierarchy of nonlinear equations

Abstract

We deduce the symplectic form for the Hamiltonian structure of a new class of nonlinear equations with the help of square eigenfunctions associated with the corresponding linear problem. The method actually yields two pieces of information simultaneously. One is the structure of the square eigenfunctions, which is of prime importance in the study of the inverse problem. The other is the form of the symplectic structure fixing up the canonical Poisson bracket relation. Finally we discuss some reductions of the initial system and the corresponding change of the Hamiltonian structure and the form of the square eigenfunction.