60 research outputs found
Abelian Functions for Cyclic Trigonal Curves of Genus Four
We discuss the theory of generalized Weierstrass and functions
defined on a trigonal curve of genus four, following earlier work on the genus
three case. The specific example of the "purely trigonal" (or "cyclic
trigonal") curve is discussed in detail, including a list of some of the associated
partial differential equations satisfied by the functions, and the
derivation of an addition formulae.Comment: 23 page
Abelian Functions for Trigonal Curves of Genus Three
We develop the theory of generalized Weierstrass sigma- and \wp-functions
defined on a trigonal curve of genus three. In particular we give a list of the
associated partial differential equations satisfied by the \wp-functions, a
proof that the coefficients of the power series expansion of the sigma-function
are polynomials of moduli parameters, and the derivation of two addition
formulae.Comment: 32 pages, no figures. Revised version has the a fuller description of
the general (3,4) trigonal curve results, the first version described only
the "Purely Trigonal" cas
Closed Loop Solitons and Sigma Functions: Classical and Quantized Elasticas with Genera One and Two
Closed loop solitons in a plane, whose curvatures obey the modified
Korteweg-de Vries equation, were investigated. It was shown that their
tangential vectors are expressed by ratio of Weierstrass sigma functions for
genus one case and ratio of Baker's sigma functions for the genus two case.
This study is closely related to classical and quantized elastica problems.Comment: AMS-Tex Use 12 page
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