A generalized Zakharov–Shabat equation with finite-band solutions and a soliton-equation hierarchy with an arbitrary parameter

Abstract

In this paper, a generalized Zakharov–Shabat equation ( g-ZS equation), which is an isospectral problem, is introduced by using a loop algebra G ∼ . From the stationary zero curvature equation we define the Lenard gradients { g j } and the corresponding generalized AKNS ( g-AKNS) vector fields { X j } and X k flows. Employing the nonlinearization method, we obtain the generalized Zhakharov–Shabat Bargmann ( g-ZS-B) system and prove that it is Liouville integrable by introducing elliptic coordinates and evolution equations. The explicit relations of the X k flows and the polynomial integrals { H k } are established. Finally, we obtain the finite-band solutions of the g-ZS equation via the Abel–Jacobian coordinates. In addition, a soliton hierarchy and its Hamiltonian structure with an arbitrary parameter k are derived.