Elliptic solutions of the Korteweg-de Vries equation

Abstract

in which the dependence of xj on t was obtained by a direct substitution of a corresponding ansatz into the KdV equation. In 1977 in [4] a relationship was uncovered between the dynamics of poles xi(t) of special solutions u(x, t) of the KdV equation (u(x, 0) = g(g + l)P(x), Lame potentials [2]) and Hamiltonian systems of particles on a line with Hamiltonians (N = g(g + 1)/2) ,u = t/2 - 2 P Soon after [i] appeared, A. R. Its and V. B. Matveev found an expression for finitezoned solutions of the KdV equation of genus g of general position in terms of g-dimensional theta functions of Riemann of a hyperelliptic curve, branch points of which are extremities of the zone spectrum for the associated Schrodinger operator. And practically at the same time the question arose as to the place of special elliptic solutions of equations of KdV type among all finite-zoned solutions, expressed in terms of multidimensional theta functions of Riemann. Interest in the problem of reducing abelian functions of genus g to elliptic functions arose for several reasons. First of all, a solution of the given problem makes it possible to describe solutions periodic in x and t, and, in a number of oases, to clarify the relationship between various ansatzes. Secondly, an expression for solutions of integrable nonlinear equations in terms of elliptic functions makes their study somewhat easier. In particular, we note that the theory of perturbations of Bogolyubov, Mitropol'skii, and Whitham [6, 7] for multizone solutions of nonlinear equations, developed for completely integrable systems by H. Flashka, G. Forest, D. McLaughlin [8], S. Yu. Dobrokhotov and V. P. Maslov [9], and B. A. Dubrovin and S. P. Novikov [i0], in which as the zero approximation use was made of multizone solutions reduced to elliptic functions, leads to simpler equations for sluggish variables than in the case of the general situation. To the present time, several methods have been worked out for reducing the many-dimensional theta functions of Riemann to theta functions of lesser dimensions including even elliptic functions. E. D. Belokolos and V. Z. Enol'skii developed a program for obtaining elliptic solutions from general finite-zoned solutions using a reduction of abelian integrals to elliptic integrals [11-13]. However, on account of the labor-consuming calculations concrete formulas were obtained only for two-zoned solutions. Another approach to this problem, developed by M. V. Babich, A. I. Bobenko, and V. B. Matveev [13-15], and subsequently by the author [16-18], was based on properties of Riemann surfaces with nontrivial groups of birational automorphisms.