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

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

1994-01-18

arxiv.org/abs/hep-th/9401084v4

Phys.Lett. B327 (1994) 59-66

Physics

High Energy Physics

High Energy Physics - Theory

10 pages, LaTex, SISSA--ISAS 10/94/EP (Revised version: the reference [12] added; small changes in the Introduction)

Scientific paper

10.1016/0370-2693(94)91529-6

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.

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