The Riemann theta function near soliton limit

Source 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 $θ$-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