On the Algebraic--Geometrical Solutions of the sine--Gordon Equation

TL;DR

This paper explores the relationship between two classes of theta function solutions to the sine-Gordon equation, revealing their equivalence and geometric connections via hyperelliptic Riemann surfaces and automorphisms.

Contribution

It demonstrates that two previously known solution classes of the sine-Gordon equation are actually the same, linking their geometric origins through coverings of Riemann surfaces.

Findings

The two classes of solutions coincide.

Solutions are related via double unramified coverings.

Discussion of soliton limits of the solutions.

Abstract

We examine the relation between two known classes of solutions of the sine--Gordon equation, which are expressed by theta functions on hyperelliptic Riemann surfaces. The first one is a consequence of the Fay's trisecant identity. The second class exists only for odd genus hyperelliptic Riemann surfaces which admit a fixed--point--free automorphism of order two. We show that these two classes of solutions coincide. The hyperelliptic surfaces corresponding to the second class appear to be double unramified coverings of the Riemann surfaces corresponding to the first class of solutions. We also discuss the soliton limits of these solutions.