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

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 {gj} and the corresponding generalized AKNS (g-AKNS) vector fields {Xj} and Xk 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 Xk flows and the polynomial integrals {Hk} 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.

► A generalized Zakharov–Shabat equation is obtained.
► The generalized AKNS vector fields are established.
► The finite-band solution of the g-ZS equation is obtained.
► By using a Lie algebra presented in the paper, a new soliton hierarchy with an arbitrary parameter is worked out.