TY - JOUR
T1 - A generalized Zakharov-Shabat equation with finite-band solutions and a soliton-equation hierarchy with an arbitrary parameter
AU - Zhang, Yufeng
AU - Tam, Honwah
AU - Feng, Binlu
N1 - Funding Information:
This work was supported by the Natural Science Foundation of Shandong Province ( ZR2009AL021 ).
PY - 2011/11
Y1 - 2011/11
N2 - 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 {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.
AB - 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 {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.
UR - http://www.scopus.com/inward/record.url?scp=80054035332&partnerID=8YFLogxK
U2 - 10.1016/j.chaos.2011.07.014
DO - 10.1016/j.chaos.2011.07.014
M3 - Journal article
AN - SCOPUS:80054035332
SN - 0960-0779
VL - 44
SP - 968
EP - 976
JO - Chaos, Solitons and Fractals
JF - Chaos, Solitons and Fractals
IS - 11
ER -