Periodic Multi-Soliton Solutions of Korteweg-de Vries Equation and Toda Lattice

Abstract

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