Genus 4 trigonal reduction of the Benney equations

Abstract

, the present authors have constructed examples of such reductions where the mappings take the upper half p-plane to a polygonal slit domain in the λ-plane. In those cases the mapping function was expressed in terms of the derivatives of Kleinian σ functions of hyperelliptic curves, restricted to the 1-dimensional stratum Θ 1 of the Θ-divisor. This was done using an extension of the method given in Enolskii et al (2003 J. Nonlinear Sci. 13 157) extended to a genus 3 curve (V Z Enolski and J Gibbons, Addition theorems on the strata of the theta divisor of genus three hyperelliptic curves, (in preparation)). Here, we use similar ideas, but now applied to a trigonal curve of genus 4. Fundamental to this approach is a family of differential relations which σ satisfies on the divisor. Again, it is shown that the mapping function is expressible in terms of quotients of derivatives of σ on the divisor Θ 1. One significant by-product is an expansion of the leading terms of the Taylor series of σ for the given family of (3, 5) curves; to the best of the authors' knowledge, this is new.

Abelian functions for cyclic trigonal curves of genus 4

“…A series expansion of the σ-function for the (3,5) curve was found by two of the present authors in [10], and that series, extended to higher order, plays a crucial role in some of the proofs below. They also found explicit formulae for a basis of differentials and for the Jacobi inversion formula for the curve.…”

“…The following formula involving the fundamental Kleinian ℘-functions was derived in [9] and was evaluated for the case of the curve (2.3) in [10]. Here it is noted that the unique Klein bidifferential may be written in two different ways, either in terms of the second derivatives of ln(σ), or else in terms of rational functions of the coordinates of points on the curve.…”

Abelian Functions for Trigonal Curves of Genus Three

“…The above derivation of the fundamental 2-form is done in [1], around page 194, and it was reconsidered in [16] for a large family of algebraic curves. The case of a trigonal curve of genus four was developed in [5], pp. 3617-3618.…”

Higher Genus Abelian Functions Associated with Cyclic Trigonal Curves

“…5. Denote the corresponding Schur-Weierstrass polynomial by SW n,s and derive it by making the change of variables p w i = w i u g+1−i , i = 1, .…”

“…differentials on this surface. We can derive the basis of holomorphic differentials using the Weierstrass gap-sequence (see for example [5])…”