A Periodicity Problem for the Korteweg–de Vries and Sturm–liouville Equations. Their Connection with Algebraic Geometry

1. As was shown in the remarkable communication [4] the Cauchy problem for the Korteweg–de Vries (KdV) equation ut = 6uux−uxxx, familiar in theory of nonlinear waves, is closely linked with a study of the spectral properties of the Sturm–Liouville operator Lψ = Eψ, where L = −d/dx + u(x). For rapidly decreasing initial conditions u(x, 0), where ∫∞ −∞ u(x, 0)(1 + |x|)dx < ∞, the KdV equation can in a sense be completely solved by going over to scattering data for the opreator L; these data determine the potential u, as was shown in [3], [8], [10]. It was later pointed out in [7] that this procedure actually provides the basis for representing the operator of multiplication by the right side of the KdV equation in terms of the commutator [A,L] = (6uux − uxxx), where A = −4d/dx + 3(u d/dx + (d/dx)u). It follows from this that the KdV equation is equivalent to the equation L̇ = [A,L]. In the case of functions u(x, t) periodic in x, and even more in the case of conditionally periodic functions, it has not proved possible to make any serious use of the connection between the operator L and the KdV equation. Recently, in [2], [9], the present authors made substantial progress in this problem, and discovered deep links with algebraic geometry. It should be mentioned that a substantial part of Dubrovin’s results [2] was obtained simultaneously and independently by Matveev and Its [6], Both the articles [2], [6] employed an idea of N. I. Ahiezer [l], the full significance of which has only recently been appreciated.