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

Yufeng Zhang, Honwah Tam and Binlu Feng

Chaos, Solitons & Fractals, 2011, vol. 44, issue 11, 968-976

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 {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.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:44:y:2011:i:11:p:968-976

DOI: 10.1016/j.chaos.2011.07.014

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