Decomposition and straightening out of the coupled mixed nonlinear Schrödinger flows

The nonlinearization approach of Lax pairs is extended to the investigation of a soliton hierarchy proposed by Wadati, Konno and Ichikawa, in which the first nontrivial equation is the coupled mixed nonlinear Schrödinger equation. Under a constraint between the potentials and eigenfunctions, solutions of the soliton hierarchy are decomposed into a class of new finite-dimensional Hamiltonian systems. The generating function of the integrals of motion is presented, by which the class of finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. Based on the decomposition and the theory of algebraic curve, the Abel–Jacobi coordinates are introduced to straighten out the corresponding flows. As an application, the compatible solutions of the various flows in Abel–Jacobi coordinates are explicitly obtained.