发表机构
The Ohio State University（俄亥俄州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明 Riemann theta 函数在孤子极限附近可近似为 Hirota 型孤子解之和，由此导出空间镶嵌，并应用于 KdV/KP 方程的准周期解构造及准周期背景上的孤子研究。
AI 中文摘要
已知雅可比椭圆函数的平方可以表示为形状为 $\sech^2$ 的单个孤子的无穷和。本文表明，对于更高亏格的情形也存在类似的结构。事实证明，这仅仅是 Riemann $\theta$-函数的准周期性的一个推论。更精确地说，我们证明了在孤子极限附近，亏格 $g$ 的实 Riemann $\theta$-函数的 $\log \theta$ 的二阶导数可以很好地近似为 $g$ 维实空间 $\R^g$ 中 Hirota 型实正则 $g$-孤子解的和。这导致 $\R^g$ 的一种镶嵌，其瓦片是一个斜棱柱，被 theta 函数中主导指数的超平面划分为 $2^g$ 个部分。我们将这些结果应用于研究 KdV 和 KP 方程的准周期解。我们使用 Schottky 群构造准周期解，该群对相应的 Riemann 曲面进行单值化。我们还通过收缩某些同调环来讨论准周期背景上的孤子。
英文摘要
It has been known that (the square of) Jacobi elliptic function can be expressed by an infinite sum of single solitons of $\sech^2$ shape. In this paper, we show that there exists a similar structure for higher genus cases. It turns out that this is just a consequence of the quasi-periodicity of the Riemann $θ$-function. More precisely, we show that the second derivative of $\log θ$ with the \emph{real} Riemann $θ$-function of genus $g$ near soliton limit can be well approximated by the \emph{sum} of \emph{real} and \emph{regular} $g$-soliton solutions of Hirota-type in $g$-dimensional real space $\R^g$. This leads to a tessellation of $\R^g$, whose tile is an oblique prism divided into $2^g$ sections by hyper-planes of dominant exponents in the theta function. We apply the results to study quasi-periodic solutions to the KdV and KP equations. We construct the quasi-periodic solutions using the Schottky group, which uniformizes the corresponding Riemann surfaces. We also discuss solitons on quasi-periodic background by pinching some of the homological cycles
Comments24 pages, 19 figures
