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Non-crossing permutations for the KP solitons under the Gel'fand-Dickey reductions and the vertex operators

Shilong Huang, Yuji Kodama, Chuanzhong Li

TL;DR

This work classifies regular KP solitons under Gel'fand-Dickey $\ell$-reductions by encoding soliton data in $A\in Gr(N,M)_{\ge0}$ and a spectral curve $\Phi_\ell(\kappa,\alpha)=\varphi_\ell(\kappa)-\alpha$, whose real roots $\kappa_j[\alpha]$ determine real exponential bases. A central result is that mutual non-crossing of the constituent blocks $A[\alpha_k]$ guarantees total nonnegativity of the combined matrix and yields a regular soliton with permutation $\pi(A[\alpha_1,...,\alpha_K])=\prod_i\pi(A[\alpha_i])$, linking algebraic, combinatorial, and analytic structures. The Boussinesq equation (3-reduction) is used to illustrate that regular solitons comprise two counterpropagating line-soliton families and at most one resonant Y-soliton, with explicit amplitude-velocity relations ensuring regularity. A vertex-operator construction for the $\ell$-reductions further shows that non-crossing block configurations produce regular solitons, connecting $Gr(N,M)_{\ge0}$ geometry with operator methods. Together, these results advance understanding of regular KP solitons, their spectral data, and potential bidirectional soliton-gas interpretations.

Abstract

We give a classification of the $regular$ soliton solutions of the KP hierarchy, referred to as the $KP solitons$, under the Gel'fand-Dickey $\ell$-reductions in terms of the permutation of the symmetric group. As an example, we show that the regular soliton solutions of the (good) Boussinesq equation as the 3-reduction can have $at ~most$ one resonant soliton in addition to two sets of solitons propagating in opposite directions. We also give a systematic construction of these soliton solutions for the $\ell$-reductions using the vertex operators. In particular, we show that the $non-crossing$ permutation gives the regularity condition for the soliton solutions.

Non-crossing permutations for the KP solitons under the Gel'fand-Dickey reductions and the vertex operators

TL;DR

This work classifies regular KP solitons under Gel'fand-Dickey -reductions by encoding soliton data in and a spectral curve , whose real roots determine real exponential bases. A central result is that mutual non-crossing of the constituent blocks guarantees total nonnegativity of the combined matrix and yields a regular soliton with permutation , linking algebraic, combinatorial, and analytic structures. The Boussinesq equation (3-reduction) is used to illustrate that regular solitons comprise two counterpropagating line-soliton families and at most one resonant Y-soliton, with explicit amplitude-velocity relations ensuring regularity. A vertex-operator construction for the -reductions further shows that non-crossing block configurations produce regular solitons, connecting geometry with operator methods. Together, these results advance understanding of regular KP solitons, their spectral data, and potential bidirectional soliton-gas interpretations.

Abstract

We give a classification of the soliton solutions of the KP hierarchy, referred to as the , under the Gel'fand-Dickey -reductions in terms of the permutation of the symmetric group. As an example, we show that the regular soliton solutions of the (good) Boussinesq equation as the 3-reduction can have one resonant soliton in addition to two sets of solitons propagating in opposite directions. We also give a systematic construction of these soliton solutions for the -reductions using the vertex operators. In particular, we show that the permutation gives the regularity condition for the soliton solutions.
Paper Structure (17 sections, 12 theorems, 133 equations, 6 figures)

This paper contains 17 sections, 12 theorems, 133 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. The KP hierarchy
  4. $N$ truncation and the $\tau$-function
  5. KP solitons
  6. The Gel'fand-Dickey $\ell$-reductions
  7. Spectral curves and the $\tau$-functions for the KP solitons under the $\ell$-reductions
  8. The spectral curve and the $\tau$-function
  9. Non-crossing permutations and the main theorem
  10. Regular solitons for the Boussinesq equation
  11. The Boussinesq equation from the KP theory
  12. One soliton solution of the Boussinesq equation
  13. Multi-soliton solutions
  14. Discussions
  15. Vertex operator construction of the KP solitons under the $\ell$-reductions
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $\{i_1,\ldots,i_N\}$ be the pivot set and $\{j_1,\ldots,j_{M-N}\}$ be the nonpivot set of $A\in\text{Gr}(N,M)_{\ge0}$. Then there exists a unique derangement $\pi$ of the symmetric group $S_M$ associated with the matrix $A$, denoted by $\pi(A)\in S_M$, so that the KP soliton has the following as

Figures (6)

  • Figure 1: The spectral curve $\Phi_7(\kappa,\alpha)=0$ with the roots $\kappa_j[\alpha_i]$ for $i=1,2$.
  • Figure 2: The left figure is the elliptic curve given by \ref{['eq:Belliptic']}, and the right one is the spectral curve $\kappa^3-c_1\kappa=\alpha$.
  • Figure 3: For each $\alpha=\alpha_j$, there are three roots labeled as $(\kappa_1[\alpha_j]<\kappa_2[\alpha_j]<\kappa_3[\alpha_j])$.
  • Figure 4: The chord diagrams $\pi(A[\alpha])$ for 1-soliton and Y-solitons.
  • Figure :
  • ...and 1 more figures

Theorems & Definitions (27)

  • Remark 1.1
  • Theorem 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • Proposition 3.4
  • proof
  • Lemma 3.5
  • ...and 17 more