A hierarchy of generalized AKNS equations, N-Hamiltonian structures and finite-dimensional involutive systems and integrable systems

Abstract

An eigenvalue problem and the associated hierarchy of nonlinear soliton equations are proposed in this paper. In particular, a representative system of generalized AKNS soliton equations in the hierarchy is given, namely, qt=−12 qxx+q2r−2μq2rx−2μrqqx+2μ2q3r2, rt=12 rxx−qr2−2μr2qx−2μqrrx+2μ2q2r3. N-Hamiltonian structures are also established for all the hirarchy of generalized AKNS soliton equations based on N+1 pairs of Hamiltonian pairs. And then the eigenvalue problem is nonlinear as a finite-dimensional completely integrable Hamiltonian system under the Bargmann constraint between the potentials and the eigenvalue functions, and its involutive system is also given. Finally, the involutive solutions of the hierarchy of generalized AKNS soliton equations are found, in particular, the involutive solutions of the system of generalized Schrödinger equations are given.