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

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.

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

References in corpus (6)

• Quantum Solitons in Affine Toda Field Theories
• The Sine-Gordon Solitons as a N-Body Problem
• Hirota's Solitons in the Affine and the Conformal Affine Toda Models
• Affine Solitons: A Relation Between Tau Functions, Dressing and Bäcklund Transformations
• Connection between the Affine and Conformal Affine Toda Models and their Hirota's Solution
• Isospectral flow in Loop Algebras and Quasiperiodic Solutions of the Sine-Gordon Equation

Cited by in corpus (1)

• Dressing Transformations and the Algebraic--Geometrical Solutions in the Conformal Affine $sl(2)$ Toda Model