, 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.
Abstract. 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 3 = x 5 + λ 4 x 4 + λ 3 x 3 + λ 2 x 2 + λ 1 x + λ 0 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.
Abstract. 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 N -parameter families of conformal maps. Here, we consider a specific set of these, the hyperelliptic reductions. The mapping function for this is calculated explicitly by inverting a second-kind Abelian integral on the stratum Θ 1 of the Jacobi variety of a genus g (g ≥ 3) hyperelliptic curve. This is done using a method based on the result of Jorgenson (1992 Israel Journal of Mathematics 77 273).
Abstract. We consider N -parameter reductions of the Benney moment equations. These were shown in Gibbons and Tsarev (1996 Phys. Lett. A 211 19, 1999 Phys. Lett. A 258 263 ) to correspond to N −parameter families of conformal maps and to satisfy a particular system of PDE. A specific known example of this, the (N = 2) elliptic reduction (L Yu and J Gibbons 2000 Inverse Problems 16 605 ) is described. We then consider an analogous reduction for a genus 2 hyperelliptic curve (N = 3). The mapping function is given by the inversion of a 2nd kind Abelian integral on the Θ−divisor. This is found explicitly following a method given by Enolskii, Pronine and Richter (2003 J. Nonlinear Science 13 157).
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