Integrability of Nonlinear Hamiltonian Systems by Inverse Scattering Method

A simple and direct scheme is presented to test the integrability of nonlinear evolution equations by inverse scattering method. The time part of the Lax equation needed for inverse scattering transform is identified with the linearized equation of the original nonlinear Hamiltonian system, while the Lax spectral operator is identified with a recursion operator connecting polynomial solutions of the linearized equation. This spectral operator is obtained by using a perturbative linear mode coupling scheme. A simple example discovered to be integrable by our scheme is shown explicitly to illustrate the detail procedures.