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

Author

Listed:
  • Zhang, Yufeng
  • Tam, Honwah
  • Feng, Binlu

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.

Suggested Citation

  • Zhang, Yufeng & Tam, Honwah & Feng, Binlu, 2011. "A generalized Zakharov–Shabat equation with finite-band solutions and a soliton-equation hierarchy with an arbitrary parameter," Chaos, Solitons & Fractals, Elsevier, vol. 44(11), pages 968-976.
    • Handle: RePEc:eee:chsofr:v:44:y:2011:i:11:p:968-976
    DOI: 10.1016/j.chaos.2011.07.014
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