A hierarchy of generalized AKNS equations, N-Hamiltonian structures and finite-dimensional involutive systems and integrable systems
A hierarchy of generalized AKNS equations, N-Hamiltonian structures and finite-dimensional involutive systems and integrable systems
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DOI：
10.1063/1.1289379
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发表时间：
2001-01
影响因子：
1.3
通讯作者：
Zhenya Yan;Hong-qing Zhang
中科院分区：
文献类型：
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作者：
Zhenya Yan;Hong-qing Zhang
本文提出了一类非线性孤子方程的本征值问题及其族。特别地，给出了该族中一个有代表性的广义AKNS孤子方程组，即：qt=−12 qxx+ q2 r −2μ q2 rx −2μrqqx+2μ 2 q3 r2，rt=12 rxx− qr 2 −2μ r2 qx −2μqrrx+2μ 2 q2 r3.对于所有的基于N+1对Hamilton对的广义AKNS孤子方程，也建立了N-Hamilton结构。然后将特征值问题作为一个有限维完全可积的Hamilton系统，在势函数和特征值函数之间的Bargmann约束下进行非线性处理，并给出了其对合系统。最后，求出了广义AKNS孤子方程族的对合解，特别是给出了广义Schrodinger方程组的对合解。
An eigenvalue problem and the associated hierarchy of nonlinear soliton equations are proposed in this paper. In particular, a representative system of generalized AKNS soliton equations in the hierarchy is given, namely, qt=−12 qxx+q2r−2μq2rx−2μrqqx+2μ2q3r2, rt=12 rxx−qr2−2μr2qx−2μqrrx+2μ2q2r3. N-Hamiltonian structures are also established for all the hirarchy of generalized AKNS soliton equations based on N+1 pairs of Hamiltonian pairs. And then the eigenvalue problem is nonlinear as a finite-dimensional completely integrable Hamiltonian system under the Bargmann constraint between the potentials and the eigenvalue functions, and its involutive system is also given. Finally, the involutive solutions of the hierarchy of generalized AKNS soliton equations are found, in particular, the involutive solutions of the system of generalized Schrodinger equations are given.