Dispersion Relations for Nonlinear Waves and Schottky Problem

An approach to the Schottky problem of specification of periods of holomorphic differentials on Riemann surfaces (or, equivalently, specification of Jacobians among all principaly polarized Abelian varieties) based on the theory of Kadomtsev Petviashvili equation, is discussed. Introduction. Dispersion relations for linear and nonlinear waves. One of the first exercises in a course of PDE is in finding particular solutions. For linear PDE the simplest solutions can be found immediately using wellknown properties of the exponential. For example, for linear wave (or Helmholtz) equation utt − uxx + m u = 0 (0.1) one can try to find a solution of the form u(x, t) = Ae. (0.2) Here A, k, ω are unknown parameters. After substitution in the equation one obtains a constraint for the parameters ω, k ω − k = m (0.3) and no constraints for the amplitude A because of linearity of the equation. The solution (0.2) is called plane wave, or one-phase solution of (0.1). The parameters A, k, ω are the amplitude, the wave number† and the frequency of the plane wave. The equation (0.3) thus is the dispersion relation for the plane waves. The solution is 2π k -periodic in x and 2π ω -periodic in t for real ω, k. The solution (0.2) is a complex one; to obtain a real solution one can take the real part of (0.2). Multiphase quasi-periodic solutions of (0.1) are linear superpositions of plane waves