Genus 4 trigonal reduction of the Benney equations

It was shown by Gibbons and Tsarev (1996 Phys. Lett. A 211 19, 1999 Phys. Lett. A 258 263) that N-parameter reductions of the Benney equations correspond to particular N-parameter families of conformal maps. In recent papers (Baldwin and Gibbons 2003 J. Phys. A: Math. Gen. 36 8393-417, Baldwin and Gibbons 2004 J. Phys. A: Math. Gen. 37 5341-54), 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 X-plane. In those cases, the mapping function was expressed in terms of the derivatives of Kleinian or functions of hyperelliptic curves, restricted to the one-dimensional stratum Theta(1) of the Theta-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 (Enolski V Z and Gibbons J 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 sigma satisfies on the divisor. Again, it is shown that the mapping function is expressible in terms of quotients of derivatives of sigma on the divisor Theta(1). One significant by-product is an expansion of the leading terms of the Taylor series of a for the given family of (3, 5) curves; to the best of the authors' knowledge, this is new.