The Riemann theta function near soliton limit

Abstract

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 $\theta$-function. More precisely, we show that the second derivative of $\log \theta$ with the \emph{real} Riemann $\theta$-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 ${\mathbb{R}}^g$. This leads to a tessellation of ${\mathbb{R}}^g$, 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

24 pages, 19 figures