|Soliton equations (some nonlinear PDEs with solitary like solutions) can be solved by the inverse scattering transform (IST) method. And the nonlinearization of the Lax pairs of some soliton equations (like the KdV equation and the AKNS hierarchy) gives us some finite dimensional integrable systems. The solutions to soliton equations can be obtained by solving the finite dimensional systems. Unfortunately, this method works equation by equation, and the search for a single method that works for every soliton equation is meaningful.
This paper finds a new method which works for every soliton equation. We use the general Legendre transformation to restrict the infinite dimensional integrable Hamiltonian systems on an invariant submanifold and obtain the finite dimensional completely integrable system. As a special example, the AKNS hierarchy is discussed.