Genus 4 trigonal reduction of the Benney equations
Genus 4 trigonal reduction of the Benney equations
复制标题
Benney 方程的属 4 三角约简
DOI：
10.1088/0305-4470/39/14/008
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发表时间：
2006
期刊：
影响因子：
--
通讯作者：
J. Gibbons
中科院分区：
文献类型：
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作者：
S. Baldwin;J. Gibbons
Gibbons 和 Tsarev (1996 Phys. Lett. A 211 19, 1999 Phys. Lett. A 258 263) 表明 Benney 方程的 N 参数约简对应于共形映射的特定 N 参数族。在最近的论文中（Baldwin and Gibbons 2003 J. Phys. A: Math. Gen. 36 8393–417，Baldwin and Gibbons 2004 J. Phys. A: Math. Gen. 37 5341–54），作者构建了此类简化的示例，其中映射将上半 p 平面映射到 λ 平面中的多边形狭缝域。在这些情况下，映射函数以超椭圆曲线的 Kleinian σ 函数的导数表示，仅限于 θ 除数的一维层 θ1。这是使用 Enolskii 等人 (2003 J. Nonlinear Sci. 13 157) 中给出的方法的扩展来完成的，该方法扩展到 genus 3 曲线（Enolski V Z 和 Gibbons J 加法定理关于 genus 3 超椭圆曲线的 theta 除数的层，正在准备中）。在这里，我们使用类似的想法，但现在应用于 4 格的三角曲线。这种方法的基础是 σ 在除数上满足的微分关系族。再次表明，映射函数可以用 σ 对除数 θ1 的导数的商来表示。一个重要的副产品是给定 (3, 5) 曲线族的 σ 泰勒级数的主要项的展开；据作者所知，这是新的。
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 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 one-dimensional stratum Θ1 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 (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 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 Θ1. 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.