Completely integrable Hamiltonian systems associated with matrix operators and Abelian varieties
Completely integrable Hamiltonian systems associated with matrix operators and Abelian varieties
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DOI：
10.1007/bf01077141
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发表时间：
1977
影响因子：
0.4
通讯作者：
B. Dubrovin
中科院分区：
文献类型：
--
作者：
B. Dubrovin
本文用推广Novikov，Matveev，及其（见综述[i]）方法的方法，研究了与一阶矩阵线性微分算子相关的非线性系统周期问题的积分。具有物理意义的这类系统的例子有非线性薛定谔方程[2]、非线性介质中波包相互作用方程[3]和修正的Korteweg-de弗里斯方程（复）。Manakov [4]已利用本文所得结果解决了刚体运动经典欧拉问题的n维推广（Mishchenko和Fomenko [5]对[4]中所得积分的独立性给出了直接的证明），给出了构造这类非线性系统的一般算法，Zakharov和Shabat [6]提出了一种求解相应线性算子反问题（在快速下降的情况下）的方法。我们给出了另一个算法，这是更方便的周期问题的积分。研究的基本对象是特征函数在有限亏格的黎曼曲面上是亚纯的矩阵算子。这些称为有限区域算子，曲面本身称为谱。这些运营商的系数满足汉密尔顿系统的固定Korteweg-德弗里斯（KdV）方程（诺维科夫方程）的类型的常微分方程。这样，积分这个系统的问题就与谱理论的逆问题，即求所有具有给定谱的有限区域算子的问题同时得到了解决。其基本结果是：具有给定谱的有限区域算子集（在关于交换群作用的因子分解内）是相应黎曼曲面的雅可比簇。对所考虑的非线性偏微分方程的时间动力学进行了完整的计算。矩阵算子的系数的显式公式中找到的0-函数。
This paper deals with the integration of a periodic problem for nonlinear systems associated with matrix linear differential operators of first order by methods that generalize the methods of Novikov, the author, Matveev, and Its (see survey [i]). Examples of such systems that are of physical interest are the nonlinear Schr~ dinger equation [2], the equation of interact $ on of wave packets in nonlinear media [3], and the modified Korteweg--de Vries equation (complex). Manakov [4] has solved the n-dimensional generalization of the classical Euler problem of the motion of a rigid body by using the results obtained in this paper (a direct verification of the independence of the integrals obtained in [4] was given by Mishchenko and Fomenko [5]).A gener~ l algorithm of construction of such nonlinear systems, together with a method of solution of the inverse problem (in the case of rapid decrease) for the corresponding linear operators, was found by Zakharov and Shabat [6]. We give another algorithm that is more convenient for the integration of the periodic problem. The basic objects of the investigation are matrix operators whose characteristic functions are meromorphic on a Riemann surface of finite genus. These are called finite-zone operators, and the surface itself is called the spectrum. The coefficients of these operators satisfy a Hamiltonian system of ordinary differential equations of the type of the stationary Korteweg--de Vries (KdV) equations (the Novikov equations). The problem of integrating this system is thus solved simultaneously with the inverse problem of spectral theory, ie, the problem of findin~ all the finite-zone operators with given spectrum. The basic result is that the set of finitezone operators with given spectrum is (to within a factorization with respect to the action of a commutative group) the Jacobian variety of the corresponding Riemann surface. The temporal dynamics for the nonlinear partial differential equations under consideration are completely calculated. Explicit formulas for the coefficients of the matrix operators found are given in terms of 0-functions.