Theta functions and non-linear equations

Abstract

CONTENTS Introduction Chapter I. Theta functions. General information § 1. Definition of theta functions and their simplest properties § 2. Theta functions of a single variable § 3. On Abelian tori § 4. Addition theorems for theta functions Chapter II. Theta functions of Riemann surfaces. The Jacobi inversion problem § 1. Periods of Abelian differentials on Riemann surfaces. Jacobi varieties § 2. Abel's theorem § 3. Some remarks on divisors on a Riemann surface § 4. The Jacobi inversion problem. Examples Chapter III. The Baker-Akhiezer function. Applications to non-linear equations § 1. The Baker-Akhiezer one-point function. The Kadomtsev-Petviashvili equation and equations associated with it § 2. The Baker-Akhiezer two-point function. The Schrödinger equation in a magnetic field Chapter IV. Effectivization of the formulae for the solution of KdV and KP equations. Recovery of a Riemann surface from its Jacobi variety. The problem of Riemann and the conjecture of Novikov § 1. The KdV equation. Genus or 2 § 2. The KP equation. Genus 2 and 3 § 3. The KP equation. Genus . Canonical equations of Riemann surfaces § 4. The problem of Riemann on relations between the periods of holomorphic differentials on a Riemann surface and the conjecture of Novikov Chapter V. Examples of Hamiltonian systems that are integrable in terms of two-dimensional theta functions § 1. Two-zone potentials § 2. The problem of Sophie Kovalevskaya § 3. The problems of Neumann and Jacobi. The general Garnier system § 4. Movement of a solid in an ideal fluid. Integration of the Clebsch case. A multi-dimensional solid Appendix. I.M. Krichever. The periodic non-Abelian Toda chain and its two-dimensional generalization References