Algebraic-geometric principles of superposition of finite-zone solutions of integrable non-linear equations

Abstract

CONTENTS Introduction Chapter I. Reduction of Abelian integrals and theta functions § 1. Abelian integrals and Riemann theta functions § 2. Reduction of Abelian integrals and theta functions of genus 2 § 3. Normal coverings and the reduction of theta functions Chapter II. Multiphase (finite-zone) solutions, expressed by Jacobi theta functions, of non-linear equations of KdV-type of genus § 4. Solutions of the “sine-Gordon” equation by elliptic functions § 5. Two-zone Lamé potentials and the associated reduction of hyperelliptic integrals § 6. On a periodic solution of a problem of Kovalevskaya § 7. Solutions of the Landau-Lifschitz equation References