ON SINGULAR SOLITONS OF THE KP EQUATION AND THE GO-DIAGRAMS

Q: How are KP soliton solutions parameterized in this paper?

Each KP soliton solution is determined by spectral data (κ, A), where κ is an ordered set of spectral parameters and A is a full-rank matrix. The associated τ-function is expressed through Plücker coordinates indexed by N-element subsets.

Q: What combinatorial objects correspond to regular versus singular solutions?

Regular KP soliton solutions are described by Γ-diagrams from Postnikov’s stratification of the totally nonnegative Grassmannian. For the singular solutions arising from failed regularity conditions, the paper parametrizes them using Go-diagrams from the Deodhar decomposition.

Q: Why do products of vertex operators often produce singular solutions?

The paper analyzes combinations of regular line-solitons via products of vertex operators, where an interaction (cross-ratio) factor depends on the relative ordering of indices. When the indices form the crossing pattern, the resulting solution becomes singular.

Oct 26, 2026

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Real and regular soliton solutions of the KP equation are classified via the totally nonnegative Grassmannian, and vertex operators generate many such solutions. This work studies regular solutions and their combinations formed by products of vertex operators, showing that generic combinations become singular. The paper proves that the resulting singular solutions admit combinatorial parametrizations by Go-diagrams, extending Postnikov’s Γ-diagrams and arising in the Deodhar decomposition.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages20

Reading Time50 min

AccessFree to download

Contents

1. Introduction

2. Grassmannian description of KP soliton solutions

3. Vertex operator construction and κ-direct sum

4. The Deodhar decomposition of the Grassmannian and Go-diagrams

5. Examples

References

arXiv : 2606 .04663v2 [nlin .SI] 23 Sep 2026 ON SINGULAR SOLITONS OF THE KP EQUATION AND THE GO-DIAGRAMS SHILONG HUANG Abstract. It has been proven that real and regular soliton solutions of the KP equation are classified in terms of the totally nonnegative Grassmannian. It is well known that vertex operators can be used to construct soliton solutions. In this paper, we consider several regular soliton solutions and study their combinations through products of vertex operators. In general, the resulting solutions become singular. Totally nonnegative elements are parametrized by the Γ-diagrams introduced by Postnikov. We show that the resulting singular solutions can be parametrized by Go-diagrams, which extend Γ-diagrams and arise in the Deodhar decomposition of the Grassmannian. Contents 1. Introduction 1 2. Grassmannian description of KP soliton solutions 4 3. Vertex operator construction and κ-direct sum 5 3.1. The vertex operators for totally nonnegative matrices 6 3.2. Products of vertex operators V [Ak ] and κ-direct sums 6 4. The Deodhar decomposition of the Grassmannian and Go-diagrams 9 4.1. Deodhar decomposition 10 4.2. Deodhar components and the associated permutations 11 4.3. Composite diagrams of the κ-direct sum 12 4.4. Go-diagrams via pipe dream 14 5. Examples 19 5.1. A case of Gr(3 , 7) 19 5.2. Hirota N-solitons 19 References 20 1. Introduction The Kadomtsev-Petviashvili (KP) equation (1.1) (−4ut + 6uux + uxxx)x + 3uyy = 0 admits soliton solutions expressed in terms of a τ-function (1.2) u (x, y, t) = 2(lnτ(x, y, t))xx . Each KP soliton solution is determined by a pair of data (κ, A), where κ = (κ1 < κ2 < ··· < κM ) is an ordered set of spectral parameters and A is an N × M full rank matrix. The associated τ-function can be expressed as τ (x, y, t) = P ∆I (A)EI (x, y, t), I Date: September 24, 2026 . 2 SHILONG HUANG where I runs over all N-element subsets of [M] := {1,..., M}, ∆I (A) denotes the Pl¨ucker coordinate indexed by I, namely the corresponding maximal minor of A, and EI (x, y, t) =Y(κik − κil )Ei1 ···EiN , Ej (x, y, t) = eξj (x,y,t) = eκjx+κ2jy+κ3jt . k>l This gives a parametrization of KP soliton solutions by the Grassmannian Gr(N, M), i.e. the space of N-dimensional subspaces of RM [2, 3] . It is known in [11] that a KP soliton solution is real and regular if and only if A ∈ Gr(N, M)≥0 , where Gr(N, M)≥0 denotes the totally nonnegative Grassmannian, i.e. all Pl¨ucker coordinates of A are nonnegative. As shown in [2, 10], the asymptotic structure of such solutions is encoded by a permutation π (A) ∈ SM , which determines the pairing of line-solitons. These line-solitons are labeled by pairs (i, j) with i < j, and are called line-solitons of type [i, j] . Regular KP soliton solutions admit a combinatorial description via Γ -diagrams, namely decorated Young diagrams with certain boxes filled by circles, which parametrize the positroid cells in Postnikov’s stratification of the totally nonnegative Grassmannian [14] . The rows and columns of a Γ-diagram are labeled by the pivot and nonpivot indices of A, respectively, so that each box is indexed by a pair (i, j) with i < j. Moreover, these diagrams satisfy the Γ -property: if a box contains a circle, then all boxes weakly to its left or above also contain circles [11] . However, singular KP soliton solutions are much less understood. For the KdV equation, i.e. the 2-reduction of the KP hierarchy, combinations of regular one-soliton solutions are always regular. Surprisingly, for the Boussinesq equation corresponding to the 3-reduction, combining two regular onesoliton solutions can produce a singular solution [7] . This is the first case where total nonnegativity is lost under combinations of regular soliton solutions. As shown in [7], vertex operators can be used to construct regular soliton solutions, and their combinations are realized by products of vertex operators. In that wo

FAQs

How are KP soliton solutions parameterized in this paper?

Each KP soliton solution is determined by spectral data (κ, A), where κ is an ordered set of spectral parameters and A is a full-rank matrix. The associated τ-function is expressed through Plücker coordinates indexed by N-element subsets.

What combinatorial objects correspond to regular versus singular solutions?

Regular KP soliton solutions are described by Γ-diagrams from Postnikov’s stratification of the totally nonnegative Grassmannian. For the singular solutions arising from failed regularity conditions, the paper parametrizes them using Go-diagrams from the Deodhar decomposition.

Why do products of vertex operators often produce singular solutions?

The paper analyzes combinations of regular line-solitons via products of vertex operators, where an interaction (cross-ratio) factor depends on the relative ordering of indices. When the indices form the crossing pattern, the resulting solution becomes singular.

Oct 26, 2026

Upload by Rowan