一族耦合的KdV方程及其对应的有限维可积系统
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摘要
1990年，曹策问先生在上海非线性物理国际学术会议上做了题为“通过特征值问题的非线性化产生经典可积系统”的报告，在这篇报告中他提出了非线性化方法。这种方法开始应用到很多2×2矩阵谱问题，得到了数十个有限维可积系统。后来，有人把这种方法推广应用到三阶和四阶高阶矩阵谱问题。高阶矩阵谱问题是物理界很感兴趣的问题，是目前国际上孤子理论研究的趋势。但由于这种问题计算量大、计算复杂，对高阶矩阵谱问题非线性化的研究较少。
当我们考虑高阶的矩阵谱问题时，需要一些特殊的技巧和进行大量的计算。特别地，体现在算子对的获得和证明守恒积分的对合性和独立性上。
本文考虑了一个有三个位势的4×4矩阵谱问题：导出一族新的非线性演化方程，其中一个典型的方程是耦合KdV方程，它在物理学中有很重要的应用。同时，借助迹恒等式，这族方程还具有广义双Hamiltonian结构。通过Bargmann约束我们得到一个有限维Hamiltonian系统。利用驻定零曲率方程解矩阵V(λ)的特征多项式：可得到2N个对合的守恒积分。文中用母函数方法给出对合性的证明，从而证明了该Hamiltonian系统在Liouville意义下是完全可积的。此外，还得到了耦合KdV方程的对合解。
By introducing a 4x4 matrix spectral problem with three potentials, we derive a new hierarchy of nonlinear evolution equations. A typical equation in the hierarchy is a coupled KdV equation. It is shown that the hierarchy possesses the generalized bi-Hamiltonian structures with the aid of the trace identity. Through the nonlinearization of eigenvalue problems, we get a new finite-dimensional Hamiltonian system, which is completely integrable in Liouville sense. In the end, we obtain the involutive solution of the coupled KdV equation.
当我们考虑高阶的矩阵谱问题时，需要一些特殊的技巧和进行大量的计算。特别地，体现在算子对的获得和证明守恒积分的对合性和独立性上。
本文考虑了一个有三个位势的4×4矩阵谱问题：导出一族新的非线性演化方程，其中一个典型的方程是耦合KdV方程，它在物理学中有很重要的应用。同时，借助迹恒等式，这族方程还具有广义双Hamiltonian结构。通过Bargmann约束我们得到一个有限维Hamiltonian系统。利用驻定零曲率方程解矩阵V(λ)的特征多项式：可得到2N个对合的守恒积分。文中用母函数方法给出对合性的证明，从而证明了该Hamiltonian系统在Liouville意义下是完全可积的。此外，还得到了耦合KdV方程的对合解。
By introducing a 4x4 matrix spectral problem with three potentials, we derive a new hierarchy of nonlinear evolution equations. A typical equation in the hierarchy is a coupled KdV equation. It is shown that the hierarchy possesses the generalized bi-Hamiltonian structures with the aid of the trace identity. Through the nonlinearization of eigenvalue problems, we get a new finite-dimensional Hamiltonian system, which is completely integrable in Liouville sense. In the end, we obtain the involutive solution of the coupled KdV equation.
引文
[1] Cao C W,Geng X G.Classical integrable systems generated through nonlinearization of eigenvalue problems.Nonlinear Physics (Research Report in Physics) (Berlin: Springer),1990,pp:68-78．
[2] Cao C W.Nonlinearization of the Lax system for the AKNS hierarchy.Sci.China,1990, A 33(5) :528-536．
[3] Cao C W,Geng X G.Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy.J.Phys.A:Math.Gen,1990,23:4117-4125．
[4] Du D L.New approach to generation of finite-dimensional integrable systems.J. Zhengzhou University,1997,29(1) :1-7．
[5] Ma W X,Geng X G.New completely integrable Neumann systems related to the perturbation KdV hierarchy.Phys.Lett.B,2000,475:56-62．
[6] Shi Q Y,Zhu S M.A finite-dimensional integrable system and the r-matrix method.J. Math.Phys,2000,41(4) :2157-2166．
[7] Wu Y T,Geng X G.A finite-dimensional integrable system associated with the three-wave interaction equations.J.Math.Phys 1999,40:3409-3430．
[8] Cao C.W.J.Math.Phys,1990,40:3948．
[9] Li M R.The trace formula of a matrix differential operator with application of integrable system.Phys.Lett.A,2001,288:196-206．
[10] Li X M.A new finite-dimensional integrable system associated with the generalized Schr dinger equation.IL NUOVO CIMENTO SECT.B,2001,116(4) :459-474．
[11] Tu G Z.The trace identity,a powerful tool for constructing the Hamiltonain structure of integrable systems.1． Math.Phys,1989,30(2) :330-338．
[12] Li M R.Some trace formulas for the eigenvalue problem of Schr dinger system.Journal of Physical Society of Japan,1999,68(4) :1107-1114．
[13] Wu Y T,Geng X G,Hu X B,Zhu S M.A generalized Hirota-Satsuma coupled Korteweg-de Vries equation and Miura transformation.Phys.Lett.A,1999,255: 259-264．
[14] Arnold V I.Mathematical Methods in Classical Mechanics,1978,(Berlin:Springer).
[2] Cao C W.Nonlinearization of the Lax system for the AKNS hierarchy.Sci.China,1990, A 33(5) :528-536．
[3] Cao C W,Geng X G.Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy.J.Phys.A:Math.Gen,1990,23:4117-4125．
[4] Du D L.New approach to generation of finite-dimensional integrable systems.J. Zhengzhou University,1997,29(1) :1-7．
[5] Ma W X,Geng X G.New completely integrable Neumann systems related to the perturbation KdV hierarchy.Phys.Lett.B,2000,475:56-62．
[6] Shi Q Y,Zhu S M.A finite-dimensional integrable system and the r-matrix method.J. Math.Phys,2000,41(4) :2157-2166．
[7] Wu Y T,Geng X G.A finite-dimensional integrable system associated with the three-wave interaction equations.J.Math.Phys 1999,40:3409-3430．
[8] Cao C.W.J.Math.Phys,1990,40:3948．
[9] Li M R.The trace formula of a matrix differential operator with application of integrable system.Phys.Lett.A,2001,288:196-206．
[10] Li X M.A new finite-dimensional integrable system associated with the generalized Schr dinger equation.IL NUOVO CIMENTO SECT.B,2001,116(4) :459-474．
[11] Tu G Z.The trace identity,a powerful tool for constructing the Hamiltonain structure of integrable systems.1． Math.Phys,1989,30(2) :330-338．
[12] Li M R.Some trace formulas for the eigenvalue problem of Schr dinger system.Journal of Physical Society of Japan,1999,68(4) :1107-1114．
[13] Wu Y T,Geng X G,Hu X B,Zhu S M.A generalized Hirota-Satsuma coupled Korteweg-de Vries equation and Miura transformation.Phys.Lett.A,1999,255: 259-264．
[14] Arnold V I.Mathematical Methods in Classical Mechanics,1978,(Berlin:Springer).