Fonctions m\'eromorphes et fonctions th\^eta sur les surfaces de Riemann

TL;DR

This paper provides an analytical overview of meromorphic theta functions on compact Riemann surfaces, illustrating their applications in integrable systems through classical examples like Kirchhoff, Landau-Lifshitz, and sine-Gordon equations.

Contribution

It offers a concise exposition of meromorphic theta functions using Mumford's framework, highlighting their role in solving integrable systems.

Findings

Application to Kirchhoff equations in Clebsch and Lyapunov-Steklov cases

Analysis of Landau-Lifshitz equation solutions

Insights into sine-Gordon equation via theta functions

Abstract

Theta functions play a major role in many current researches and are powerful tools for studying integrable systems. The purpose of this paper is to provide a short and quick exposition of some aspects of meromorphic theta functions for compact Riemann surfaces. The study of theta functions will be done via an analytical approach using meromorphic functions in the framework of Mumford. Some interesting examples will be given : the classical Kirchhoff equations in the cases of Clebsch and Lyapunov-Steklov, the Landau-Lifshitz equation and the sine-Gordon equation.