A generalized Zakharov–Shabat equation with finite-band solutions and a soliton-equation hierarchy with an arbitrary parameter
A generalized Zakharov–Shabat equation with finite-band solutions and a soliton-equation hierarchy with an arbitrary parameter
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DOI：
10.1016/j.chaos.2011.07.014
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发表时间：
2011-11
影响因子：
7.8
通讯作者：
Yufeng Zhang;H. Tam;B. Feng
中科院分区：
文献类型：
--
作者：
Yufeng Zhang;H. Tam;B. Feng
本文利用圈代数G引入了一个广义Zakharov-Shabat方程（g-ZS方程），它是一个等谱问题.从定常零曲率方程定义了Lenard梯度{gj}和相应的广义AKNS（g-AKNS）向量场{Xj}和Xk流。利用非线性化方法得到了广义Zhakharov-Shabat Bargmann（g-ZS-B）系统，并通过引入椭圆坐标和发展方程证明了它是Liouville可积的.建立了Xk流与多项式积分{Hk}的显式关系。最后，我们通过Abel-Jacobian坐标得到了g-ZS方程的有限带解。此外，还导出了一个孤子族及其具有任意参数k的哈密顿结构。
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 Xkflows. 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 Xkflows 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.