Non-Linear Equations of Korteweg-De Vries Type, Finite-Zone Linear Operators, and Abelian Varieties

The basic concept of this survey is an exposition of a recently developed method of constructing a broad class of periodic and almost-periodic solutions of non-linear equations of mathematical physics to which (in the rapidly decreasing case) the method of the inverse scattering problem is applicable. These solutions are such that the spectrum of their associated linear differential operators has a finite-zone structure. The set of linear operators with a given finite-zone structure is the Jacobian variety of a Riemann surface, which is determined by the structure of the spectrum. We give an explicit so-lution of the corresponding non-linear equations in the language of the theory