Periodic Multi-Soliton Solutions of Korteweg-de Vries Equation and Toda Lattice
Periodic Multi-Soliton Solutions of Korteweg-de Vries Equation and Toda Lattice
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DOI：
10.1143/ptps.59.107
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发表时间：
1976
影响因子：
--
通讯作者：
E. Date;Shun’ichi Tanaka
中科院分区：
文献类型：
--
作者：
E. Date;Shun’ichi Tanaka
本文综述了近年来关于KdV方程周期多孤子解的研究工作。强调了与阿贝尔积分理论的联系。我们还讨论了离散类比，得到了周期Toda晶格的精确解。Gardner、Greene、Kruskal和Miura(GG KM)的作品ll和lax。2>；利用逆散射理论，GG Km发现多孤子解--经典孤立波解的适当推广--可以由无反射势构造出来。经过Novikov3和GT的工作，建立了多孤子解的周期模拟与谱中具有有限个间隙的势相联系。这些势和KdV方程的相关解的显式实现由Dubrovin4l和ITS-Matveev给出。他们的方法是基于阿贝尔积分的理论，并回到Akhiezer.对这些著作进行回顾是本文的目的所在。作为一条独立的发展线，Lax8>和McKean-Moerbeke9从Hochstadt10l的早期工作出发，利用哈密顿形式研究了高阶KdV方程的周期问题。文中还指出了与阿贝尔积分理论的关系。11)。在§2中，我们描述了具有周期势的Sturm-Liouville方程的谱性质的一般性。研究了谱中具有有限个数(如g)带隙的势。在§3中，引入了亏格g的超椭圆Riemann曲面，使得Bloch特征函数是曲面上的单值函数。然后
A review of the recent works on the periodic multi-soliton solutions of the KdV equation is given. Connection with the theory of the abelian integrals is emphasized. Discrete analogue is also discussed leading to the exact solutions of the periodic Toda lattice. the works of Gardner, Greene, Kruskal and Miura (GG KM)ll and Lax. 2> On employing the inverse scattering theory, GG KM have found that the multi-soliton solutions, the proper generalization of the classical solitary wave solutions, can be constructed from the reflectionless potentials. After the work of Novikov3> it is established that the periodic analogue of the multi-soliton solutions is connected with the potentials which have finite number of gaps in the spectrum. An explicit realization of these potentials and associated solutions of the KdV equation is given by Dubrovin4l and Its-Matveev.5l Their method is based on the theory ·of the abelian integrals and goes back to Akhiezer.6l The present authors7> develop a discrete analogue of the construction of Dubrovin and Its-Matveev. To give a review of these works is the purpose of the present paper. As an independent line of development, Lax8> and then McKean Moerbeke9l have studied the periodic problem for the Sturm-Liouville equation and the KdV equation starting from the earlier work of Hochstadt10l and employing the Hamiltonian formalism for the higher order KdV equations. The relation with the theory of the abelian integrals is also indicated in Ref. 11). In §2 we describe generalities on the spectral properties of the Sturm Liouville equation with periodic potential. The potentials with finite number (say g) of gaps in the spectrum are studied. In §3 a hyperelliptic Riemann surface of genus g is introduced so that the Bloch eigenfunction is single-valued function on the surface. Then