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KP solitons and the Riemann theta functions

Yuji Kodama

TL;DR

The paper demonstrates that regular KP solitons arising from the totally nonnegative Grassmannian can be captured by degenerate Riemann theta functions on singular curves, via the $M$-theta function with parameters explicitly tied to the soliton data. It establishes a precise correspondence between $\tau$-functions and theta-function data, connects to Grammian/ Pfaffian representations, and discusses Prym theta functions and quasi-periodic backgrounds realized through vertex operators. The framework is illustrated with concrete $\mathrm{Gr}(N,M)$ examples, including degenerate and resonant interactions, and extended to solitons on quasi-periodic backgrounds, highlighting a unifying algebro-geometric view of KP solitons and their interactions.

Abstract

We show that the $τ$-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.

KP solitons and the Riemann theta functions

TL;DR

The paper demonstrates that regular KP solitons arising from the totally nonnegative Grassmannian can be captured by degenerate Riemann theta functions on singular curves, via the -theta function with parameters explicitly tied to the soliton data. It establishes a precise correspondence between -functions and theta-function data, connects to Grammian/ Pfaffian representations, and discusses Prym theta functions and quasi-periodic backgrounds realized through vertex operators. The framework is illustrated with concrete examples, including degenerate and resonant interactions, and extended to solitons on quasi-periodic backgrounds, highlighting a unifying algebro-geometric view of KP solitons and their interactions.

Abstract

We show that the -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.
Paper Structure (12 sections, 6 theorems, 108 equations)

This paper contains 12 sections, 6 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. The Riemann theta function
  3. The Riemann theta function on a singular curve
  4. The Prym theta function on a singular curve
  5. The KP equation
  6. Quasi-periodic solutions
  7. Soliton solutions
  8. The $M$-theta function as the $\tau$-function of the KP solitons
  9. Examples
  10. Hirota type soliton solutions
  11. A degenerate example
  12. KP solitons on a quasi-periodic background

Key Result

Lemma 2.1

For $j\ne k$, we have where $:\hat{V}_j\cdot\hat{V}_k:$ is the normal ordering of the product $\hat{V}_j\cdot\hat{V}_k$, and we have used eq:Omega.

Theorems & Definitions (21)

  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Definition 3.2
  • Remark 3.3
  • Definition 3.4
  • Lemma 3.5
  • proof
  • Example 3.6
  • ...and 11 more