KP solitons and the Riemann theta functions

TL;DR

This paper establishes a connection between KP soliton solutions and Riemann theta functions on singular curves, providing explicit parameter relations and exploring quasi-periodic backgrounds.

Contribution

It demonstrates that KP soliton tau-functions can be expressed via Riemann theta functions on singular curves, linking soliton theory with algebraic geometry.

Findings

KP solitons from totally nonnegative Grassmannians relate to Riemann theta functions.

Explicit formulas connect KP soliton parameters with Riemann theta function parameters.

Discussion of quasi-periodic KP solitons via vertex operators and theta functions.

Abstract

We show that the $\tau$-functions of the regular KP solitons from the totally nonnegative Grassmannians can be expressed by the Riemann theta functions on singular curves. We explicitly write the parameters in the Riemann theta function in terms of those of the KP soliton. We give a short remark on the Prym theta function on a double covering of singular curves. We also discuss the KP soliton on quasi-periodic background, which is obtained by applying the vertex operators to the Riemann theta function.