The Theory of Binary Nonlinearization and Its Applications to Soliton Equations

Binary symmetry constraints are applied to the nonlinearization of spectral problems and adjoint spectral problems into so-called binary constrained flows, which provide candidates for finite-dimensional Liouville integrable Hamiltonian systems. The resulting constraints on the potentials of spectral problems give rise to a kind of involutive solutions to zero curvature equations, and thus the integrability by quadratures can be shown for zero curvature equations once the corresponding binary constrained flows are proved to be integrable. The whole process to carry out binary symmetry constraints is called binary nonlinearization. The principal task of binary nonlinearization is to expose the Liouville integrability for the resulting binary constrained flows, which can usually be achieved as a consequence of the existence of hereditary recursion operators. The theory of binary nonlinearization is applied to the multi-wave interaction equations associated with a 4 × 4 matrix spectral problem as an illustrative example. The Bäcklund transformations resulted from symmetry constraints are given for the multi-wave interaction equations, and thus a kind of involutive solutions is presented and the integrability by quadratures is shown for the multi-wave interaction equations.