Abelian Functions Associated with a Cyclic Tetragonal Curve of Genus Six
نویسنده
چکیده
We develop the theory of Abelian functions defined using a tetragonal curve of genus six, with the specific example of the cyclic curve, y = x + λ4x + λ3x + λ2x + λ1x + λ0 discussed in detail. We define generalisations of the Weierstrass σ and ℘ functions, along with additional classes of Abelian functions. In addition, we present the associated partial differential equations satisfied by the functions, the solution of the Jacobi Inversion Problem, a power series expansion for σ(u) and an addition formula. Finally we demonstrate how such Abelian functions can generate solutions to the KPequation.
منابع مشابه
ar X iv : 0 80 6 . 23 77 v 2 [ m at h . A G ] 3 N ov 2 00 8 Abelian functions associated with a cyclic tetragonal curve of genus six
We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve y 4 = x 5 + λ 4 x 4 + λ 3 x 3 + λ 2 x 2 + λ 1 x + λ 0. We construct Abelian functions using the multivariate σ-function associated to the curve, generalising the theory of the Weierstrass ℘-function. We demonstrate that such functions can give a solution to the KP-equa...
متن کاملA genus six cyclic tetragonal reduction of the Benney equations
A reduction of Benney's equations is constructed corresponding to Schwartz-Christoffel maps associated with a family of genus six cyclic tetragonal curves. The mapping function, a second kind Abelian integral on the associated Riemann surface, is constructed explicitly as a rational expression in derivatives of the Kleinian σ-function of the curve.
متن کاملHigher Genus Abelian Functions Associated with Cyclic Trigonal Curves
We develop the theory of Abelian functions associated with cyclic trigonal curves by considering two new cases. We investigate curves of genus six and seven and consider whether it is the trigonal nature or the genus which dictates certain areas of the theory. We present solutions to the Jacobi inversion problem, sets of relations between the Abelian function, links to the Boussinesq equation a...
متن کاملAbelian Functions for Cyclic Trigonal Curves of Genus Four
We discuss the theory of generalized Weierstrass σ 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 y = x + λ4x 4 + λ3x 3 + λ2x 2 + λ1x+ λ0 is discussed in detail, including a list of some of the associated partial differential equations satisfied by the ℘ functio...
متن کاملAbelian Functions for Purely Trigonal Curves of Genus Three
We develop the theory of generalized Weierstrass σ and ℘ functions defined on a trigonal curve of genus three. The specific example of the “purely trigonal”curve y = x +λ3x 3 +λ2x 2 +λ1x+λ0 is discussed in detail, including a list of the associated partial differential equations satisfied by the ℘ functions, and the derivation of two addition formulae.
متن کامل