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 sech2 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 Rg. This leads to a tessellation of Rg, whose tile is an oblique prism divided into 2g 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
Related
- Proof of the Riemann Hypothesis through the limit of an approximated ξ function
- Convergence in law of the maximum of the Riemann zeta function on expanding intervals near the critical line
- Poisson-Dirichlet statistics for the log-modulus of the Riemann zeta function on expanding intervals near the critical line
- Small values of the Riemann zeta-function near the critical line
- Geometric Proofs of Riemann-Type Theta Relations
BibTeX
@misc{kodama2026riemann,
title = {{The Riemann theta function near soliton limit}},
author = {Yuji Kodama},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05716},
eprint = {2610.05716},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05716},
}
RIS
TY - UNPB
TI - The Riemann theta function near soliton limit
AU - Yuji Kodama
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.05716
UR - https://arxiv.org/abs/2610.05716
AB - 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
ER -