Finite-gap solutions of the Sine-Gordon equation

TL;DR

This paper presents the initial results on finite-gap integration of the Sine-Gordon equation, providing explicit solutions expressed via theta-functions, extending methods used for the Korteweg-de Vries equation to this nonlinear wave equation.

Contribution

It introduces the finite-gap solutions of the Sine-Gordon equation and provides explicit theta-function formulae, adapting techniques from integrable systems theory.

Findings

Explicit finite-gap solutions in terms of theta-functions.

Methodology for constructing solutions analogous to KdV equation.

Original results from 1976 translated into English for wider accessibility.

Abstract

This paper contains first results on the finite-gap integration of the Sine-Gordon equation. They were published on Russian in 1976. The papers \cite{Koz}, \cite{KK}, \cite{KK02} have been rewritten in the English language with small modifications for a convenience. Such a translation was made due to requests of some interested readers. In those papers, the method of constructing of the finite-gap solutions of the equation $u_{tt}-u_{xx}+\sin u=0$ was proposed. The explicit formulae were obtained for these solutions. The formulae are constructed in terms of $\theta$-functions and they are analogous to the formulae obtained by A.R.Its and V.B.Matveev \cite{IM}, B.A.Dubrovin and S.P.Novikov \cite{DN} for periodic and almost periodic solutions to the Korteweg de Vries equation.