KP solitons and the Riemann 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.

1 Introduction

It is well known that the KP solitons are given by the so-called $\tau$-function in the form of Wronskian determinant (see, e.g., [12] and below). The regular KP solitons have been classified in terms of the totally nonnegative Grassmannian $\text {Gr}(N,M)$ where N and M are positive integers with $N<M$ [5, 19]. These solitons are uniquely determined by a set of distinct parameters $(\kappa _1,\kappa _2,\ldots ,\kappa _{M})\in {\mathbb {R}}^M$ and an element $A\in \text {Gr}(N,M)$ (see, e.g., [16, 18]). It is also known that the quasi-periodic (algebro-geometric) solutions of the KP equation can be constructed by the Riemann theta function on smooth curve (see, e.g., [2]). The real regular solutions are also discussed in [2].

In this paper, we show that the regular KP solutions can be expressed by the theta function on singular curve with several degenerations of double points. The theta function on singular curve with ordinary double points was obtained by Mumford in his famous book [23]. We call this theta function the M-theta function in this paper. Then, we give explicit formulas of the M-theta function in terms of $(\kappa _1,\ldots ,\kappa _M)$ and $A\in \text {Gr}(N,M)$. It turns out that a general KP soliton is obtained by a further degeneration of the nodal points. It is quite interesting to study these singular structure from a viewpoint of algebraic geometry. But we will not discuss the details in this note. We would like to mention a recent paper [14], where the singular structure of the Riemann surface has been studied in connection with the $\tau$-function of the KP solutions [14]. We also give a remark on the Prym theta function as a $\tau$-function for the BKP hierarchy, which is given by a Pfaffian form [11, 21, 29].

The paper is organized as follows. In Sect. 2, we give a brief description of the Riemann theta function on singular curve following Chapter 5 in the book [23]. We define the M-theta function and note that the M-theta function can be expressed in the form with vertex operators. In Sect. 3, we start with a brief introduction of the regular soliton solutions of the KP equation. The section will be concluded with the main theorem (Theorem 3.10), which gives explicit formula of the $\tau$-function in terms of the M-theta function. We consider two main examples in Sect. 4, where we explicitly demonstrate how one can construct the regular KP soliton directly from the matrix $A\in \text {Gr}(N,M)$. In the final Sect. 5, we give a formula which describes the KP solitons on quasi-periodic background. The formula is obtained by applying the vertex operators given in Sect. 2.

2 The Riemann theta function

The Riemann theta function of genus g is defined as follows. Let ${\mathcal {H}}_g$ be the set of symmetric $g\times g$ matrix whose imaginary part is positive definite, called the Siegel upper-half-plane and the element $\Omega \in {\mathcal {H}}_g$ is called the period matrix. The Riemann theta function of genus g is then defined by

$$\begin{aligned} \vartheta _g(\textbf{z};\Omega ):=\mathop {\textstyle \sum }\limits _{\textbf{m}\in {\mathbb {Z}}^g}\exp 2\pi i \left( \frac{1}{2}{} \textbf{m}^T\Omega \textbf{m}+\textbf{m}^T \textbf{z}\right) , \end{aligned}$$

(2.1)

for $\textbf{z}\in {\mathbb {C}}^g$, and ${\textbf{m}}^T$ is the transpose of the column vector ${\textbf{m}}$. The $g\times g$ period matrix $\Omega$ is calculated as follows (see, e.g., [8, 22]). Let ${\mathcal {C}}$ be a compact Riemann surface of genus g. Let $\{a_1,\ldots ,a_g, b_1,\ldots ,b_g\}$ be a canonical homology basis on ${\mathcal {C}}$. Let $\{\omega _1,\ldots ,\omega _g\}$ be a basis of the normalized holomorphic differentials on ${\mathcal {C}}$. Then, the period matrix $\Omega =(\Omega _{i,j})$ is calculated by

$$\begin{aligned} \oint _{a_i}\omega _j=\delta _{i,j},\qquad \oint _{b_i}\omega _j=\Omega _{i,j},\qquad \text {for}\quad 1\le i,j\le g. \end{aligned}$$

For example of the case of $g=1$, we consider an elliptic curve $\mu ^2+\prod _{j=1}^3(\lambda -e_j)=0$ with the real parameters $e_1>e_2>e_3$. Then, we have

$$\begin{aligned} \omega =\frac{d\lambda }{A\mu } \quad \text {which gives}\quad \Omega ={\oint _b\omega }=i\frac{K'(k)}{K(k)}, \end{aligned}$$

where $A=\oint _a\frac{d\lambda }{\mu }$, $K'(k)=K(\sqrt{1-k^2})$, and K(k) is the complete elliptic integrals of the first kind,

$$\begin{aligned} K(k)=\displaystyle \int _0^1\frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}},\quad \text {with}\quad k^2=\frac{e_2-e_3}{e_1-e_3}. \end{aligned}$$

Equation (2.1) is the Jacobi theta function.

We just mention that in general, an explicit computation of the period matrix is hard for given algebraic curve (see, e.g., [10, 22, 23]).

2.1 The Riemann theta function on a singular curve

In [23][Chapter 5, p.3.243], Mumford considered the theta function on singular curve. Let C be a singular curve of (arithmetic) genus g, and let $S=\text {Sing}(C)=\{p_1,\ldots ,p_g\}\subset C$. Assume that the singularities of C are only ordinary double points $p_1,\ldots ,p_g$ and that C has normalization

$$\begin{aligned} \pi :{\mathbb {P}}~\rightarrow ~ C. \end{aligned}$$

If $\pi ^{-1}(p_i)=\{b_i,c_i\}$ for $i=1,\ldots ,g$, this implies that C is just ${\mathbb {P}}$ with g pairs of points $\{b_i,c_i\}$ identified. Then, the Riemann theta function on the singular curve C can be obtained by taking the limits of the diagonal parts of the period matrix $\Omega$,

$$\begin{aligned} \text {Im}~\Omega _{i,i}~\rightarrow ~ \infty \quad \text {for}\quad 1\le i\le g, \end{aligned}$$

(2.2)

and the limits of $\Omega _{k,l}$ for $k\ne l$ are given by

$$\begin{aligned} \Omega _{k,l}~\rightarrow ~{\tilde{\Omega }}_{k,l}:=\frac{1}{2\pi i}\ln \frac{(b_k-b_l)(c_k-c_l)}{(b_k-c_l)(c_k-b_l)}, \end{aligned}$$

(2.3)

(see also [9, 15]). The limit of (2.1) will be just 1, which corresponds to the choice $\textbf{m}^T=(0,\ldots ,0)$. To obtain a non-trivial example, we consider the shifts

$$\begin{aligned} z_i~\rightarrow ~z_i-\frac{1}{2}\Omega _{i,i},\quad \text {for}\quad i=1,\ldots ,g, \end{aligned}$$

which then gives the Riemann theta function with shifted variable $\textbf{z}=(z_1,\ldots ,z_g)^T\in {\mathbb {C}}^g$,

$$\begin{aligned} \vartheta _g(\textbf{z};\Omega )=\mathop {\textstyle \sum }\limits _{\textbf{m}\in {\mathbb {Z}}^g}\exp 2\pi i\left( \frac{1}{2}\mathop {\textstyle \sum }\limits _{i=1}^gm_i(m_i-1)\Omega _{i,i}+\mathop {\textstyle \sum }\limits _{i<j}m_im_j\Omega _{i,j}+\mathop {\textstyle \sum }\limits _{i=1}^gm_iz_i\right) ,\nonumber \\ \end{aligned}$$

(2.4)

where $\textbf{m}=(m_1,\ldots ,m_g)$. Then, the limits $\Omega _{i,i}\rightarrow +i\infty$ for all $i=1,\ldots , g$ lead to

$$\begin{aligned} \vartheta _g({\textbf{z}};\Omega )~&\rightarrow ~ ~{\tilde{\vartheta }}^{(g)}_g({\textbf{z}};{\tilde{\Omega }}):=\mathop {\textstyle \sum }\limits _{\textbf{m}\in \{0,1\}^g}\exp 2\pi i\left( \mathop {\textstyle \sum }\limits _{k<l}m_km_l{\tilde{\Omega }}_{k,l}+\mathop {\textstyle \sum }\limits _{k=1}^gm_kz_k\right) \nonumber \\&=1+\mathop {\textstyle \sum }\limits _{k=1}^g e^{2\pi i z_k}+\mathop {\textstyle \sum }\limits _{1\le k<l\le g}e^{2\pi i{\tilde{\Omega }}_{k,l}}e^{2\pi i (z_k+ z_l)}+\ldots +e^{2\pi i\mathop {\textstyle \sum }\limits _{k<l}{\tilde{\Omega }}_{k,l}}e^{2\pi i\mathop {\textstyle \sum }\limits _{j=1}^g z_j}, \end{aligned}$$

(2.5)

that is, the infinite sum of exponential terms in the $\vartheta$-function (2.1) becomes a finite sum of $2^g$ exponential terms corresponding to the case with $m_i\in \{0,1\}$. We remark here that ${\tilde{\vartheta }}({\textbf{z}};{\tilde{\Omega }})$ in (2.5) can be expressed in a simple form using the vertex operator ${{\hat{V}}}_j[{\tilde{\Omega }}]$, which is defined by

$$\begin{aligned} {\hat{V}}_j[{\tilde{\Omega }}]:=\exp 2\pi i\left( z_j+\mathop {\textstyle \sum }\limits _{k=1, k\ne j}^g{\tilde{\Omega }}_{j,k}\frac{\partial }{\partial z_k}\right) . \end{aligned}$$

(2.6)

It is easy to verify the following lemma for the product of two vertex operators.

Lemma 2.1

For $j\ne k$, we have

$$\begin{aligned} {{\hat{V}}}_j[{\tilde{\Omega }}]\cdot {{\hat{V}}}_k[{\tilde{\Omega }}]=\frac{(b_j-b_k)(c_j-c_k)}{(b_j-c_k)(c_j-b_k)}:{{\hat{V}}}_j[{\tilde{\Omega }}]\cdot {{\hat{V}}}_k[{\tilde{\Omega }}]:, \end{aligned}$$

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 (2.3).

Lemma 2.1 implies that the vertex operators commute. Then, the function ${\tilde{\vartheta }}({\textbf{z}};{\tilde{\Omega }})$ in (2.5) is expressed by

$$\begin{aligned} {\tilde{\vartheta }}^{(g)}_g({\textbf{z}};{\tilde{\Omega }})=\prod _{j=1}^g(1+{{\hat{V}}}_j[{\tilde{\Omega }}])\cdot 1. \end{aligned}$$

(2.7)

We call the ${\tilde{\vartheta }}^{(g)}_g$ function the M-theta function after Mumford (see Chapter 5 in his book [23]). We will also define ${\tilde{\vartheta }}_g^{(n)}$, which is a theta function on a singular curve of genus $g-n$ having with n singular points (see Sect. 5). In [23], the g pairs of points $\{b_i,c_i\}$ on ${\mathbb {P}}$ are taken as $b_i=-c_i$ to show that (2.7) gives a g-soliton solution of the KdV equation with appropriate choice of the variables ${\textbf{z}}$. One can also see that the M-theta function gives g-soliton solution of the KP equation (see below). In this paper, we show that any KP soliton generated by the totally nonnegative Grassmannian (see, e.g., [16]) can be written in the form (2.7) with appropriate restrictions on $z_j$ and ${\tilde{\Omega }}_{k,l}$, i.e., further limits of $\Omega _{k,l}$, which is also related to an identification with points $\{b_i,c_i\}$ on ${\mathbb {P}}$.

Remark 2.2

It is also interesting to note that the M-theta function can be expressed in the determinant form [15],

$$\begin{aligned} {\tilde{\vartheta }}_g^{(g)}({\textbf{z}};{\tilde{\Omega }})=\text {det}\left( \delta _{j,k}+\frac{b_j-c_j}{b_j-c_k}e^{\pi i(z_j+z_k)}\right) _{1\le j,k \le g} \end{aligned}$$

Expressing the variables ${\textbf{z}}$ and ${\tilde{\Omega }}$ in terms of the soliton parameters (see below), this formula can be identified as the Grammian form [6, 12].

2.2 The Prym theta function on a singular curve

Here, we give a remark on the theta function of Prym varieties of branched coverings [7, 21, 29]. First, we note that the period matrix ${\tilde{\Omega }}_{j,k}$ in (2.3) is obtained by the integral,

$$\begin{aligned} {\tilde{\Omega }}_{j,k}=\frac{1}{2\pi i}\int _{c_j}^{b_j}\,\omega _k(z), \end{aligned}$$

where $\omega _k$ is the differential of third kind (which is obtained by the holomorphic differential of the first kind in the singular limit [14, 15]),

$$\begin{aligned} \omega _k(z)=\left( \frac{1}{z-b_k}-\frac{1}{z-c_k}\right) \,dz,\qquad \text {for}\quad k=1,\ldots ,g. \end{aligned}$$

In the theory of Prym variety [21], we have a double covering map, $\mu :{\tilde{C}}\rightarrow C$ for a curve C of genus g. We also have an involution, interchanging two sheets above any point in C, $\iota :{\tilde{C}}\rightarrow {{\tilde{C}}}$ (see also [29]). The involution acts on the differentials $\omega _k^B(z)$ by

$$\begin{aligned} \iota (\omega _k^B(z))=\omega ^B_k(-z)=-\omega _k^B(z). \end{aligned}$$

Then, the normalization ${\tilde{\pi }}:{\mathbb {P}}\rightarrow {\tilde{C}}$ has the inverse image ${\tilde{\pi }}^{-1}(p^{\pm }_j)=\{\pm b_j,\pm c_j\}$ with $\mu (p^{\pm }_j)=p_j\in C$. The differentials for a singular curve C may be expressed by

$$\begin{aligned} \omega _k^B(z)=\left( \frac{1}{z-b_k}-\frac{1}{z+b_k}+\frac{1}{z-c_k}-\frac{1}{z+c_k}\right) \,dz. \end{aligned}$$

Note that one can write $\omega _k^B=\omega ^+_k+\omega _k^-$, where

$$\begin{aligned} \omega ^{\pm }_k(z)=\pm \left( \frac{1}{z{\mp } b_k}-\frac{1}{z\pm c_k}\right) \,dz. \end{aligned}$$

That is, each $\omega _k^{\pm }$ is the differential on one sheet of double covers ${\tilde{C}}$. Then, the period matrix ${\tilde{\Omega }}^B_{k,l}$ can be computed as

$$\begin{aligned} {\tilde{\Omega }}^B_{j,k}=\frac{1}{2\pi i}\int _{-c_j}^{b_j}\,\omega _k^B(z)=\frac{1}{2\pi i}\ln \frac{(b_j-b_k)(c_j-c_k)(b_j-c_k)(c_j-b_k)}{(b_j+b_k)(c_j+c_k)(b_j+c_k)(c_j+b_k)}. \end{aligned}$$

We now define the Prym M-theta function corresponding to two-sheeted coverings of singular curve, which is defined in the same form of (2.5) with ${\tilde{\Omega }}_{j,k}={\tilde{\Omega }}_{j,k}^B$,

$$\begin{aligned} {\tilde{\vartheta }}^B_g(z;{\tilde{\Omega }}^B):=\mathop {\textstyle \sum }\limits _{\textbf{m}\in \{0,1\}^g}\exp 2\pi i\left( \mathop {\textstyle \sum }\limits _{j<k}m_jm_k\,{\tilde{\Omega }}^B_{j,k}+\mathop {\textstyle \sum }\limits _{k=1}^gm_kz_k\right) . \end{aligned}$$

(2.8)

Remark 2.3

Introduce new variables $\{\xi _j:1\le j\le 2g\}$ and write

$$\begin{aligned} 2\pi i z_k=\xi _{2k-1}+\xi _{2k}+\ln \frac{b_k-c_k}{b_k+c_k},\qquad \text {for}\quad k=1,\ldots ,g. \end{aligned}$$

Then, following [11], one can show that the Prym M-theta function can be expressed by the Pfaffian of $2g\times 2g$ skew symmetric matrix,

$$\begin{aligned} {\tilde{\vartheta }}_g^B(z;{\tilde{\Omega }}^B)=\text {Pf}\left( c_{j,k}+\frac{q_j-q_k}{q_j+q_k}\,e^{\xi _j+\xi _k}\right) _{1\le j<k\le 2g}, \end{aligned}$$

(2.9)

where $c_{j,k}=-c_{k,j}$ are constants, and for $1\le j<k\le 2g$ and $1\le l\le 2g$,

$$\begin{aligned} c_{j,k}&=\left\{ \begin{array}{llll} 1,\quad &{}\text {if}~\, j=2n-1~\text {and}~k=2n,\quad (1\le n\le g),\\ 0,\quad &{}\text {otherwise} \end{array}\right. \\ q_l&=\left\{ \begin{array}{lll} b_n,\quad &{}\text {if}~\, l=2n-1,\\ c_n,\quad &{}\text {if}~\, l=2n, \end{array}\right. \quad \text {for}\quad 1\le n\le g. \end{aligned}$$

Notice that the Prym M-theta function can be also expressed using the vertex operators in the same way as (2.7) with ${\tilde{\Omega }}={\tilde{\Omega }}^B$. We will also note the Pfaffian formulas of the soliton solutions for the BKP hierarchy in the next section.

3 The KP equation

In this section, we give a basic information about the KP solitons for the purpose of the present paper (see, e.g., [16] for the details). The KP equation is a nonlinear partial differential equation in the form,

$$\begin{aligned} \partial _x(-4\partial _t u+6u\partial _xu+\partial _x^3u)+3\partial _y^2u=0, \end{aligned}$$

(3.1)

The solution of the KP equation is given in the following form,

$$\begin{aligned} u(x,y,t)=2\partial _x^2\ln \tau (x,y,t), \end{aligned}$$

(3.2)

where $\tau (x,y,t)$ is called the $\tau$-function of the KP equation.

The $\tau$-function satisfies the Hirota bilinear equation,

$$\begin{aligned} (-4D_xD_t+D_x^4+3D_y^2)\tau \circ \tau =0, \end{aligned}$$

where $D_x$’s are the Hirota derivatives defined by

$$\begin{aligned} D_x^m f\circ g:=(\partial _x-\partial _{x'})^m f(x)g(x')\Big |_{x=x'} \end{aligned}$$

3.1 Quasi-periodic solutions

A quasi-periodic solution is given by a Riemann theta function

$$\begin{aligned} \tau (x,y,t)=\vartheta _g({\textbf{u}}x+\textbf{v}y+\textbf{w}t+\textbf{z}_0,\Omega ), \end{aligned}$$

where $\textbf{u},\textbf{v}, \textbf{w}$ and $\textbf{z}_0$ are real g-dimensional constant vectors. The vectors $\textbf{u},\textbf{v}$ and $\textbf{w}$ satisfy the nonlinear dispersion relation (more precisely, the Schottky problem). The constant vectors are given by integrals of normalized meromorphic differentials (see, e.g., [2]).

In the case of $g=1$, we take

$$\begin{aligned} z=\frac{1}{4K(k)}(\alpha x+\beta y+\gamma t+\omega _0)+\frac{1}{2}, \end{aligned}$$

where the nonlinear dispersion for the constants $(\alpha , \beta , \gamma )$ is

$$\begin{aligned} \frac{-4\alpha \gamma +3\beta ^2}{\alpha ^4}=3\frac{E(k)}{K(k)}-2+k^2. \end{aligned}$$

Here E(k) is the complete elliptic integral of the second kind,

$$\begin{aligned} E(k)=\int _0^1\frac{\sqrt{1-k^2t^2}}{\sqrt{1-t^2}}\,dt,\qquad \text {for}\quad 0\le k\le 1. \end{aligned}$$

Then, the solution u(x, y, t) is given by

$$\begin{aligned} u(x,y,t)=2\partial _x^2\log \vartheta _1(z;\Omega )=2\alpha ^2\left( \text {dn}^2\frac{\alpha x+\beta y+\gamma t+\omega _0}{2}-\frac{E(k)}{K(k)}\right) . \end{aligned}$$

(3.3)

where $\text {dn}\,z$ is the Jacobi elliptic function. Then, we take the limit $k\rightarrow 1$, which is called the soliton limit, and we have

$$\begin{aligned} u(x,y,t)~\rightarrow ~2\alpha ^2\mathop {\textrm{sech}}\nolimits ^2\frac{\alpha x+\beta y+\gamma t+\omega _0}{2}. \end{aligned}$$

Here, we have used $K(k)\rightarrow \infty$ and $K'(k)\rightarrow \frac{\pi }{2}$, and note

$$\begin{aligned} \text {dn}^2 z-\frac{E}{K}=\left( \frac{\pi }{2K'}\right) ^2\mathop {\textstyle \sum }\limits _{l\in {\mathbb {Z}}}\mathop {\textrm{sech}}\nolimits ^2\frac{\pi K}{K'}\left( \frac{z}{2K}-l\right) ~\rightarrow ~\mathop {\textrm{sech}}\nolimits ^2 z. \end{aligned}$$

Note also that the dispersion relation for the soliton solution is given by

$$\begin{aligned} -4\alpha \gamma +3\beta ^2+\alpha ^4=0 \end{aligned}$$

(3.4)

Remark 3.1

In this paper, we consider the limit $k\rightarrow 0$, which gives $K'\rightarrow \infty$, i.e.,

$$\begin{aligned} \Omega ~\rightarrow ~+i\infty ,\quad \text {and}\quad K(k),~E(k)~\rightarrow ~\frac{\pi }{2}. \end{aligned}$$

This limit corresponds to the linear limit. However, considering z to be imaginary and shifting z by $\frac{\Omega }{2}$, one can also obtain the soliton solution as will be explained below. Note that changing the wave parameters $\kappa :=(\alpha ,\beta , \gamma )$ to $i\kappa$, the dispersion relation becomes (3.4).

3.2 Soliton solutions

The soliton solutions are constructed as follows: Let $\{f_i(x,y,t):1\le i\le N\}$ be a set of linearly independent functions $f_i(x,y,t)$ satisfying the following system of linear equations,

$$\begin{aligned} \frac{\partial f_i}{\partial y}=\frac{\partial ^2 f_i}{\partial x^2},\quad \text {and}\quad \frac{\partial f_i}{\partial t}=\frac{\partial ^3 f_i}{\partial x^3}\qquad i=1,\ldots ,N. \end{aligned}$$

(3.5)

The $\tau$-function is then given by the Wronskian with respect x-variable,

$$\begin{aligned} \tau (x,y,t)=\text {Wr}(f_1,f_2,\ldots ,f_N). \end{aligned}$$

(3.6)

(see, e.g., [16] for the details.)

As a fundamental set of the solutions of (3.5), we take the exponential functions,

$$\begin{aligned} E_j(x,y,t)=e^{\xi _j(x,y,t)}\quad \text {with}\quad \xi _j(x,y,t):=\kappa _jx+\kappa _j^2y+\kappa _j^3t\qquad \text {for}\quad j=1,\ldots ,M. \end{aligned}$$

where $\kappa _j$’s are arbitrary real constants. In this paper, we consider the regular soliton solutions, for which we assume the ordering

$$\begin{aligned} \kappa _1\,<\,\kappa _2\,<\,\kappa _3\,<\,\cdots \,<\,\kappa _M. \end{aligned}$$

(3.7)

For the soliton solutions, we consider $f_i(x,y,t)$ as a linear combination of the exponential solutions,

$$\begin{aligned} f_i(x,y,t)=\mathop {\textstyle \sum }\limits _{j=1}^Ma_{i,j}E_j(x,y,t)\qquad \text {for}\qquad i=1,\ldots ,N. \end{aligned}$$

(3.8)

where $A:=(a_{i,j})$ is an $N\times M$ constant matrix of full rank, $\text {rank}(A)=N$. Then, the $\tau$-function (3.6) is expressed by

$$\begin{aligned} \tau (x,y,t)=|AE(x,y,t)^T|, \end{aligned}$$

(3.9)

where $E(x,y,t)^T$ is the transpose of the matrix E(x, y, t), which is given by

$$\begin{aligned} E(x,y,t)=\begin{pmatrix} E_1 &{} E_2 &{}\cdots &{} E_M\\ \partial _1E_1&{}\partial _1E_2&{}\cdots &{}\partial _1E_M\\ \vdots &{}\vdots &{}\ddots &{}\vdots \\ \partial _1^{N-1}E_1 &{}\partial _1^{N-1}E_2&{}\cdots &{}\partial _1^{N-1}E_M \end{pmatrix}. \end{aligned}$$

(3.10)

Note here that the set of exponential functions $\{E_1,\ldots ,E_M\}$ gives a basis of M-dimensional space of the null space of the operator $\prod _{i=1}^M(\partial _x-\kappa _i)$, and we call it a “basis” of the KP soliton. Then, the set of functions $\{f_1,\ldots ,f_N\}$ represents an N-dimensional subspace of M-dimensional space spanned by the exponential functions. This leads naturally to the structure of a finite Grassmannian $\text {Gr}(N,M)$, the set of N-dimensional subspaces in ${\mathbb {R}}^M$. Then, the $N\times M$ matrix A of full rank can be identified as a point of $\text {Gr}(N,M)$, and in this paper, we assume A to be in the reduced row echelon form (RREF). Here, we consider the irreducible matrix A, which is defined as follows.

Definition 3.2

An $N\times M$ matrix A in RREF is irreducible, if

• (a) in each row, there is at least one nonzero element besides the pivot, and

• (b) there is no zero column.

This implies that the first pivot is located at (1, 1) entry, and the last pivot should be at $(N, i_N)$ with $N\le i_N<M$.

The $\tau$-function can be expressed as the following formula using the Binet–Cauchy lemma (see, e.g., [16]),

$$\begin{aligned} \tau (x,y,t)=\mathop {\textstyle \sum }\limits _{I\in \left( {\begin{array}{c}[M]\\ N\end{array}}\right) }\Delta _{I}(A)E_{I}(x,y,t), \end{aligned}$$

(3.11)

where $I=\{i_1<i_2<\cdots <i_N\}$ is an N element subset in $[M]=\{1,2,\ldots ,M\}$, $\Delta _{I}(A)$ is the $N\times N$ minor with the column vectors indexed by $I=\{i_1,\ldots ,i_N\}$, and $E_{I}(x,y,t)$ is the $N\times N$ determinant of the same set of the columns in (3.10), which is given by

$$\begin{aligned} E_{I}=\prod _{k<l}(\kappa _{i_l}-\kappa _{i_k})\,E_{i_1}\cdots E_{i_N}=\prod _{k<l}(\kappa _{i_l}-\kappa _{i_k}) \exp \left( \xi _{i_1}+\cdots +\xi _{i_N}\right) . \end{aligned}$$

(3.12)

The minor $\Delta _{I}(A)$ is also called the Plücker coordinate, and the $\tau$-function represents a point of $\text {Gr}(N,M)$ in the sense of the Plücker embedding, $\text {Gr}(N,M)\hookrightarrow {\mathbb {P}}(\wedge ^N{\mathbb {C}}^M): A\mapsto \{\Delta _I(A):I\in \left( {\begin{array}{c}[M]\\ N\end{array}}\right) \}$. It is shown in [19] that all the regular KP solitons can be obtained by choosing A from the totally nonnegative Grassmannian, where $\Delta _J(A)\ge 0$ for each index set $J=\{j_1,\ldots ,j_N\}$.

3.3 The M-theta function as the $\tau$-function of the KP solitons

Here, we give the main theorem, which shows that the $\tau$-function (3.11) can be expressed by the M-theta function with further identifications of $\{b_i,c_i\}$ on ${\mathbb {P}}$.

Let us start with the definition of the matroid for an irreducible matrix A in the totally nonnegative Grassmannian $\text {Gr}(N,M)$,

$$\begin{aligned} {\mathcal {M}}(A)=\left\{ I\in \left( {\begin{array}{c}[M]\\ N\end{array}}\right) : \Delta _I(A)>0\right\} . \end{aligned}$$

Let $I_0$ be the lexicographically minimum element of ${\mathcal {M}}(A)$. Then, we have the decomposition,

$$\begin{aligned} {\mathcal {M}}(A)=\bigcup _{n=0}^N {\mathcal {M}}_n(A), \end{aligned}$$

(3.13)

where

$$\begin{aligned} {\mathcal {M}}_n(A):=\left\{ J\in {\mathcal {M}}(A):|J\cap I_0|=N-n\right\} . \end{aligned}$$

Note that ${\mathcal {M}}_0(A)=I_0$.

Let [i, j] denote the difference of the functions $\xi _j(x,y,t)=\kappa _jx+\kappa _j^2y+\kappa _j^3t$,

$$\begin{aligned}{}[i,j]:=\xi _j-\xi _i,\quad \text {for}\quad i\in I,~j\in J, \end{aligned}$$

(3.14)

where $I,J\in {\mathcal {M}}(A)$. Notice that $(\alpha ,\beta ,\gamma )$ in $\xi _j-\xi _i=\alpha \,x+\beta \,y+\gamma \,t$ where

$$\begin{aligned} \alpha =\kappa _j-\kappa _i,\quad \beta =\kappa _j^2=\kappa _i^2,\quad \gamma =\kappa _j^3-\kappa _i^3, \end{aligned}$$

satisfies the dispersion relation of the KP equation (3.4).

Remark 3.3

The $\tau$-function defined by $\tau =1+e^\phi$ with $\phi =[i,j]$ gives one-soliton solution,

$$\begin{aligned} u(x,y,t)=2\partial _x^2\log \tau (x,y,t)=\frac{(\kappa _i-\kappa _j)^2}{2}\mathop {\textrm{sech}}\nolimits ^2\frac{1}{2}\phi (x,y,t). \end{aligned}$$

Let us define the difference $I_0-J$ for $J\in {\mathcal {M}}_1(A)$, in which each element for $J\in {\mathcal {M}}_1(A)$ is also denoted as [i, j] for $i\in I_0\setminus J$ and $j\in J\setminus I_0$. We then define the set

$$\begin{aligned} P_1(A)=\{[i,j]:i\in I_0\setminus J,~j\in J\setminus I_0,~ J\in {\mathcal {M}}_1(A)\} \end{aligned}$$

Note that each element in $P_1(A)$ can be expressed as $[i_k,j_l]$ where

• (a) $i_k\in I_0\setminus J$ is the k-th pivot of A,

• (b) $j_l\in J\setminus I_0$ is a nonzero element in the k-th row of A.

Let $j^{(k)}_l\in J\setminus I_0$ $(l=1,\ldots , n_k)$ be the nonzero elements in the kth row of A with the order $j_l^{(k)}<j_{m}^{(k)}$ for $l<m$. Then, we define

$$\begin{aligned} \phi _{{{\tilde{g}}}_{k-1}+l}:=[i_k,j_l^{(k)}]=\xi _{j_l^{(k)}}-\xi _{i_k},\quad l=1,\ldots ,n_k, \end{aligned}$$

(3.15)

where ${{\tilde{g}}}_{k}=n_1+\ldots +n_k$ with ${{\tilde{g}}}_0=0$. Let ${{\tilde{g}}}$ denote the total number of elements in ${\mathcal {M}}_1(A)$ given by

$$\begin{aligned} {{\tilde{g}}}:={{\tilde{g}}}_N=|{\mathcal {M}}_1(A)|=n_1+\cdots +n_N, \end{aligned}$$

(3.16)

which is the total number of nonzero elements in A except the pivots. Then, ${{\tilde{g}}}$ gives the number of pairs $\{\kappa _{i_k},\kappa _{j_l}\}$ in the inverse image of singular points of C, $S=\{p_1,\ldots ,p_{{{\tilde{g}}}}\}$, where we identify the points $\pi ^{-1}(p_j)=\{b_j,c_j\}$ for $1\le j\le {{\tilde{g}}}$ as

$$\begin{aligned} \{b_{{{\tilde{g}}}_{k-1}+l}=\kappa _{i_k},~ c_{{{\tilde{g}}}_{k-1}+l}=\kappa _{j_l^{(k)}}\},\quad \text {for}\quad 1\le l\le n_k~\,\text {and}~\, 1\le k\le N. \end{aligned}$$

(3.17)

One should note here that the indices of $b_i$’s are given by $I_0$, the pivot indices, and those of $c_j$’s are given by the non-pivot indices.

We also define the following sets for $2\le n\le N$. For $J=\{j_1,\ldots ,j_N\}\in {\mathcal {M}}_n(A)$,

$$\begin{aligned} P_n(A):=\left\{ \mathop {\textstyle \sum }\limits _{i_k\in I_0\setminus J}\mathop {\textstyle \sum }\limits _{j_l\in J\setminus I_0}[i_k,j_l]:[i_k,j_l]\in P_1(A),~ J\in {\mathcal {M}}_n(A)\right\} \end{aligned}$$

Note that the sets $P_n(A)$ are generated by the elements in $P_1(A)$, and each index in the sum appears only once. The number of elements $[i_k,j_l]$ in each sum in $P_n(A)$ is n, and we denote the element as

$$\begin{aligned} J-I_0=\mathop {\textstyle \sum }\limits _{i_k\in I_0\setminus J}\mathop {\textstyle \sum }\limits _{j_l\in J\setminus I_0}[i_k,j_l]. \end{aligned}$$

Note in particular that the sum is not unique, e.g., $[1,4]+[2,3]=[1,3]+[2,4]$ (see the example below). We then define the linear dependence of the elements in $P_1(A)$.

Definition 3.4

The linear independence and the dimension of $P_1(A)$.

• (a) $P_1(A)$ is said to be linearly dependent if an element $[i_k,j_l]$ in $P_1(A)$ is expressed by

  $$\begin{aligned} {[}i_k,j_l]=\mathop {\textstyle \sum }\limits \epsilon _{i,j}[i,j],\quad \text {with}~ \epsilon _{i,j}\in \{0,\pm 1\}~\text {and}~\mathop {\textstyle \sum }\limits |\epsilon _{i,j}|\ge 2, \end{aligned}$$

  where the sum is over all elements in $P_1(A)$.

• (b) The dimension of $P_1(A)$ is defined by

  $$\begin{aligned} \text {dim}\, P_1(A)=\{\text {number of independent elements in}~P_1(A)\} \end{aligned}$$

  Note that $\text {dim}\,P_1(A)\le {{\tilde{g}}}$ defined in (3.16).

Then, we have the following lemma.

Lemma 3.5

For $A\in \text {Gr}(N,M)$ (irreducible totally nonnegative element), we have

$$\begin{aligned} N\le \text {dim}\,P_1(A)\,\le M-1. \end{aligned}$$

Proof

Since A is irreducible, there is at least one $[i_k,j]$ for each row. This implies $N\le \text {dim}\,P_1(A)$. The maximum dimension is given by the case when A is in the top cell (largest dimensional cell in $\text {Gr}(N,M)$. In this case, we have $|{\mathcal {M}}_1(A)|=N(M-N)$. Each dependent relation is corresponding to the $2\times 2$ matrix,

$$\begin{aligned} \begin{pmatrix} a_{k,N+i}&{} a_{k,N+i+1}\\ a_{k+1,N+i}&{}a_{k+1,N+i+1} \end{pmatrix},\qquad \text {for}\quad 1\le i\le M-N-1~\text {and}~~1\le k\le N-1 \end{aligned}$$

that is, $[k,N+i]+[k+1,N+i+1]=[k,N+i+1]+[k+1,N+i]$, which give the total $(N-1)(M-N-1)$ independent relations. Then, we have

$$\begin{aligned} \text {dim}\,P_1(A)=N(M-N)-(N-1)(M-N-1)=M-1, \end{aligned}$$

which gives the maximum dimension. $\square$

Example 3.6

Consider the element of the top positive Grassmannian cell $\text {Gr}(2,4)$, which is

$$\begin{aligned} A=\begin{pmatrix} 1 &{}0 &{} -a &{} -b\\ 0&{}1&{}c &{} d \end{pmatrix} \end{aligned}$$

where $a,b,c,d>0$ and $cb-ad>0$. Then, we have ${\mathcal {M}}(A)={\mathcal {M}}_0(A)\cup {\mathcal {M}}_1(A)\cup {\mathcal {M}}_2(A)$ where

$$\begin{aligned} {\mathcal {M}}_0(A)=\{(1,2)\},\quad {\mathcal {M}}_1(A)=\{(1,3),~(1,4),~(2,3),~(2,4)\},\quad {\mathcal {M}}_2(A)=\{(3,4)\}. \end{aligned}$$

This gives

$$\begin{aligned} P_1(A)&=\{[1,3],~[1,4],~[2,3],~[2,4]\},\\ P_2(A)&=\{[1,3]+[2,4]=[1,4]+[2,3]\}, \end{aligned}$$

which implies that the set $P_1(A)$ is linearly dependent and $\text {dim}\,P_1(A)=3<{{\tilde{g}}}=|{\mathcal {M}}_1(A)|=4$

We write the $\tau$-function (3.11) in the following form,

$$\begin{aligned} \tau (x,y,t)&=\mathop {\textstyle \sum }\limits _{n=0}^N\mathop {\textstyle \sum }\limits _{J\in {\mathcal {M}}_n(A)}\Delta _J(A)E_J(x,y,t)\nonumber \\&=\Delta _{I_0}(A)E_{I_0}(x,y,t)\left( 1+\mathop {\textstyle \sum }\limits _{n=1}^N\mathop {\textstyle \sum }\limits _{J\in {\mathcal {M}}_n(A)}\frac{\Delta _J(A)E_J}{\Delta _{I_0}(A)E_{I_0}}\right) \end{aligned}$$

(3.18)

Since the solution is given by the second derivative of $\ln \tau$, one can take the $\tau$-function in the following form,

$$\begin{aligned} \tau (x,y,t)=1+\mathop {\textstyle \sum }\limits _{n=1}^N\mathop {\textstyle \sum }\limits _{J\in {\mathcal {M}}_n(A)}\Delta _J(A)\frac{E_J}{E_{I_0}}. \end{aligned}$$

(3.19)

where we have taken $\Delta _{I_0}(A)=1$ for the pivot set $I_0$. For each $i_k\in I_0$ and $j^{(k)}_l\in J$ of $J\in {\mathcal {M}}_1(A)$ (see (3.15) for the notation), we first note from (3.12) that

$$\begin{aligned} \frac{E_J}{E_{I_0}}=\frac{\prod _{m<n}(\kappa _{j_n}-\kappa _{j_m})}{\prod _{m<n}(\kappa _{i_n}-\kappa _{i_m})}\exp ({\xi _{j^{(k)}_l}-\xi _{i_k}}) =\frac{\prod _{m\ne k}|\kappa _{i_m}-\kappa _{j^{(k)}_l}|}{\prod _{m\ne k}|\kappa _{i_m}-\kappa _{i_k}|}\exp ({\phi _{{{\tilde{g}}}_{k-1}+l}}). \end{aligned}$$

Then, we define a constant phase parameter $\phi _j^{(0)}$ for $j=1,\ldots , {{\tilde{g}}}$,

$$\begin{aligned} \exp (\phi _{{{\tilde{g}}}_{k-1}+l}^0):=\Delta _J(A)\frac{\prod _{m\ne k}|\kappa _{i_m}-\kappa _{j^{(k)}_l}|}{\prod _{m\ne k}|\kappa _{i_m}-\kappa _{i_k}|},\quad 1\le l\le n_k. \end{aligned}$$

Note here that each nonzero entry in the matrix A is represented in the phase function $\phi _m^0$ for $1\le m\le {{\tilde{g}}}:=|{\mathcal {M}}_1(A)|$, and we have $\Delta _J(A)=|a_{k,j_l^{(k)}}|$, i.e.,

$$\begin{aligned} |a_{k,j_l^{(k)}}|=\frac{\prod _{m\ne k}|\kappa _{i_m}-\kappa _{i_k}|}{\prod _{m\ne k}|\kappa _{i_m}-\kappa _{j_l^{(k)}}|}\exp (\phi ^0_{{{\tilde{g}}}_{k-1}+l}). \end{aligned}$$

(3.20)

Then, the first term in the sum in (3.19) is given by

$$\begin{aligned} \mathop {\textstyle \sum }\limits _{J\in {\mathcal {M}}_1(A)}\Delta _J(A)\frac{E_J}{E_{I_0}}=\mathop {\textstyle \sum }\limits _{k=1}^N\mathop {\textstyle \sum }\limits _{l=1}^{n_k}\exp ({\tilde{\phi }}_{{{\tilde{g}}}_{k-1}+l})=\mathop {\textstyle \sum }\limits _{m=1}^{{{\tilde{g}}}}e^{{\tilde{\phi }}_m}, \end{aligned}$$

where ${\tilde{\phi }}_m:=\phi _m+\phi ^0_m$.

For the sum over $J\in {\mathcal {M}}_2(A)$, let $\{i_k,i_l\}\in I_0{\setminus } J$ and $\{j_m^{(k)},j_n^{(l)}\}\in J{\setminus } I_0$. Then, the following lemma is immediate

Lemma 3.7

$$\begin{aligned} \frac{E_J}{E_{I_0}}=\frac{\prod _{s\ne k,l}|\kappa _{i_s}-\kappa _{j_m^{(k)}}||\kappa _{i_s}-\kappa _{j_n^{(l)}}||\kappa _{j_m^{(k)}}-\kappa _{j_n^{(l)}}|}{\prod _{s\ne k,l}|\kappa _{i_s}-\kappa _{i_k}||\kappa _{i_s}-\kappa _{i_l}||\kappa _{i_k}-\kappa _{i_l}|}\exp ({\phi _{{{\tilde{g}}}_{k-1}+m}+\phi _{{{\tilde{g}}}_{l-1}+n}}), \end{aligned}$$

Now let $a_{k,j_m^{(k)}}$ and $a_{l,j_n^{(l)}}$ be the entries of A at the locations $(k,j_m^{(k)})$ and $(l,j_n^{(l)})$, respectively.

Proposition 3.8

$$\begin{aligned} |a_{k,j_m^{(k)}}a_{l,j_n^{(l)}}|\frac{E_J}{E_{I_0}}&=\frac{|\kappa _{i_k}-\kappa _{i_l}||\kappa _{j_m^{(k)}}-\kappa _{j_n^{(l)}}|}{|\kappa _{i_k}-\kappa _{j_n^{(l)}}||\kappa _{j_m^{(k)}}-\kappa _{i_l}|}\exp ({\tilde{\phi }}_{{{\tilde{g}}}_{k-1}+m}+{\tilde{\phi }}_{{{\tilde{g}}}_{l-1}+n}), \end{aligned}$$

where ${\tilde{\phi }}_m=\phi _m+\phi _m^0$.

Proof

Let $J_k:=(J\setminus \{j_m^{(k)}\})\cup \{i_k\}$ and $J_l:=(J\setminus \{j_n^{(l)}\})\cup \{i_l\}$. Note that $J_k,J_l\in {\mathcal {M}}_1(A)$, and we have

$$\begin{aligned} \Delta _{J_k}(A)=|a_{k,j_m^{(k)}}|,\quad \text {and}\quad \Delta _{J_l}=|a_{l,j_n^{(l)}}|. \end{aligned}$$

Then, using (3.20), we have the formula. $\square$

From (3.17), the coefficient of the exponential function gives

$$\begin{aligned} \frac{(\kappa _{i_k}-\kappa _{i_l})(\kappa _{j_m^{(k)}}-\kappa _{j_n^{(l)}})}{(\kappa _{i_k}-\kappa _{j_n^{(l)}})(\kappa _{j_m^{(k)}}-\kappa _{i_l})}=\exp \left( 2\pi i\,{\tilde{\Omega }}_{g_{k-1}+m,g_{l-1}+n}\right) \end{aligned}$$

One should note that for $\{j_m^{(k)},j_n^{(l)}\}\in J{\setminus } I_0$, we have

$$\begin{aligned} \Delta _J(A)=\Big |a_{k,j_m^{(k)}}a_{l,j_n^{(l)}}-a_{k,j_{n'}^{(k)}}a_{l,j_{m'}^{(l)}}\Big |, \end{aligned}$$

where $j_{n'}^{(k)}=j_n^{(l)}$ and $j_{m'}^{(l)}=j_m^{(k)}$. If $a_{k,j_{n'}^{(k)}}a_{l,j_{m'}^{(l)}}\ne 0$, we have the phases

$$\begin{aligned} \phi _{{{\tilde{g}}}_{k-1}+n'}=[i_k,j_{n'}^{(k)}],\quad \text {and}\quad \phi _{{{\tilde{g}}}_{l-1}+m'}=[i_l,j_{m'}^{(l)}]. \end{aligned}$$

Note here that we have a resonant condition,

$$\begin{aligned} \phi _{{{\tilde{g}}}_{k-1}+n'}+\phi _{{{\tilde{g}}}_{l-1}+m'}=\phi _{{{\tilde{g}}}_{k-1}+m}+\phi _{{{\tilde{g}}}_{l-1}+n}. \end{aligned}$$

Then, we have

$$\begin{aligned} \Delta _J(A)\frac{E_J}{E_{I_0}}&=\pm (a_{k,j_m^{(k)}}a_{l,j_n^{(l)}}-a_{k,j_{n'}^{(k)}}a_{l,j_{m'}^{(l)}})\frac{E_J}{E_{I_0}}\\&=\pm \left( \frac{(\kappa _{i_k}-\kappa _{i_l})(\kappa _{j_m^{(k)}}-\kappa _{j_n^{(l)}})}{(\kappa _{i_k}-\kappa _{j_n^{(l)}})(\kappa _{j_m^{(k)}}-\kappa _{i_l})}e^{{\tilde{\phi }}_{{{\tilde{g}}}_{k-1}+m}+{\tilde{\phi }}_{{{\tilde{g}}}_{l-1}+n}}\right. \\&\quad \left. -\frac{(\kappa _{i_k}-\kappa _{i_l})(\kappa _{j_{n'}^{(k)}}-\kappa _{j_{m'}^{(l)}})}{(\kappa _{i_k}-\kappa _{j_{m'}^{(l)}})(\kappa _{j_{n'}^{(k)}}-\kappa _{i_l})}e^{{\tilde{\phi }}_{{{\tilde{g}}}_{k-1}+n'}+{\tilde{\phi }}_{{{\tilde{g}}}_{l-1}+m'}}\right) \end{aligned}$$

Repeating the similar computations, we have the formula for $J\in {\mathcal {M}}_n(A)$.

Proposition 3.9

Let $\{i_k, i_l,\ldots ,i_m\}\in I_0{\setminus } J$ and $\{j_r^{(k)}, j_s^{(l)}, \ldots , j_t^{(m)}\}\in J{\setminus } I_0$. Then, we have

$$\begin{aligned} \left| a_{k,j_r^{(k)}}\,a_{l,j_s^{(l)}}\cdots a_{m,j_t^{(m)}}\right| \frac{E_J}{E_{I_0}}=\left| C_{{{\tilde{g}}}_{k-1}+r,\, {{\tilde{g}}}_{l-1}+s,\, \ldots , \,{{\tilde{g}}}_{m-1}+t}\right| e^{{\tilde{\phi }}_{{{\tilde{g}}}_{k-1}+r}+{\tilde{\phi }}_{{{\tilde{g}}}_{l-1}+s}+\cdots +{\tilde{\phi }}_{{{\tilde{g}}}_{m-1}+t}}, \end{aligned}$$

where $C_{k,l,\ldots ,m}$ is defined by

$$\begin{aligned} C_{k,l,\ldots ,m}=\prod _{a<b\in \{k,l,\ldots ,m\}}C_{a,b}=\exp 2\pi i\left( \mathop {\textstyle \sum }\limits _{a<b\in \{k,l,\ldots ,m\}}{\tilde{\Omega }}_{a,b}\right) , \end{aligned}$$

with

$$\begin{aligned} C_{{{\tilde{g}}}_{k-1}+r,\,{{\tilde{g}}}_{l-1}+s}:=\frac{(\kappa _{i_k}-\kappa _{i_l})(\kappa _{j_r^{(k)}}-\kappa _{j_s^{(l)}})}{(\kappa _{i_k}-\kappa _{j_s^{(l)}})(\kappa _{j_r^{(k)}}-\kappa _{i_l})}=\exp 2\pi i\,{\tilde{\Omega }}_{{{\tilde{g}}}_{k-1}+r,{{\tilde{g}}}_{l-1}+s}. \end{aligned}$$

We now state the main theorem as the summary of the results above.

Theorem 3.10

For $A\in \text {Gr}(N,M)$ (A is in the RREF), let ${{\tilde{g}}}$ be the total number of nonzero elements in A except the pivot one’s. Let $n_k$ denote the total number of nonzero elements in the k-th row, and define

$$\begin{aligned} {{\tilde{g}}}_k=n_1+\cdots +n_k,\quad (1\le k\le N),\qquad \text {and}\quad {{\tilde{g}}}={{\tilde{g}}}_N. \end{aligned}$$

Also let $j^{(k)}_m$ denote the non-pivot column index of a nonzero element in the k-th row of A (note $1\le m\le n_k$), and assume that $j_m^{(k)}<j_l^{(k)}$ if $m<l$.

Then, the $\tau$-function $\tau _A(x,y,t)$ in (3.18) can be expressed by the (degenerated) M-theta function ${\tilde{\vartheta }}_{{{\tilde{g}}}}^{({{\tilde{g}}})}({\textbf{z}};{\tilde{\Omega }})$,

$$\begin{aligned} \tau _A(x,y,t)/E_{I_0}(x,y,t)={\tilde{\vartheta }}^{({{\tilde{g}}})}_{{{\tilde{g}}}}({\textbf{z}};{\tilde{\Omega }})=\mathop {\textstyle \sum }\limits _{m\in \{0,1\}}^{{{\tilde{g}}}}\exp 2\pi i\left( \mathop {\textstyle \sum }\limits _{i<j}m_im_j{\tilde{\Omega }}_{i,j}+\mathop {\textstyle \sum }\limits _{j=1}^{{{\tilde{g}}}}m_jz_j\right) , \end{aligned}$$

where $E_{I_0}$ is the $N\times N$ determinant in (3.12) with the lexicographically minimal set $I=I_0$, ${{\tilde{g}}}={{\tilde{g}}}_N$ and $2\pi i {\textbf{z}}={\tilde{\phi }}(x,y,t)$ with ${\tilde{\phi }}_n=\phi _n+\phi ^0_n$ for $1\le n\le {{\tilde{g}}}$. We then have for $1\le k\le N$

$$\begin{aligned} \phi _{{{\tilde{g}}}_{k-1}+m}&=[i_k,j^{(k)}_m]=(\kappa _{j_m^{(k)}}-\kappa _{i_k})x+(\kappa _{j_m^{(k)}}^2-\kappa ^2_{i_k})y+(\kappa ^3_{j_m^{(k)}}-\kappa ^3_{i_k})t,\\ \exp (\phi ^0_{{{\tilde{g}}}_{k-1}+m})&=| a_{k,j_m^{(k)}}|\frac{\prod _{l\ne k}|\kappa _{i_l}-\kappa _{j^{(k)}_m}|}{\prod _{l\ne k}|\kappa _{i_l}-\kappa _{i_k}|},\\ \exp \left( 2\pi i\,{\tilde{\Omega }}_{g_{k-1}+m,g_{l-1}+n}\right)&=\frac{(\kappa _{i_k}-\kappa _{i_l})(\kappa _{j_m^{(k)}}-\kappa _{j_n^{(l)}})}{(\kappa _{i_k}-\kappa _{j_n^{(l)}})(\kappa _{j_m^{(k)}}-\kappa _{i_l})}. \end{aligned}$$

Here, $a_{k,j^{(k)}_m}$ is the element in A with the corresponding index, and assume ${{\tilde{g}}}_0=0$.

Remark 3.11

Having the variables, $\{{\tilde{\phi }}_j, {\tilde{\Omega }}_{j,k}:1\le j<k\le {{\tilde{g}}}\}$ from the matrix $A\in \text {Gr}(N,M)$ and the parameters $(\kappa _1,\ldots ,\kappa _{M})$, the $\tau$-function can be given in the form with the vertex operators (2.7). This formula has been also obtained by Nakayashiki [25], where he has constructed the $\tau$-function by taking several limits of the parameters of the vertex operators for the Hirota type solutions.

Remark 3.12

From Remark 2.2, Theorem 3.10 implies that the Wronskian form of the $\tau$-function can be also written in the Grammian form which is the M-theta function [6].

Remark 3.13

In the Pfaffian formula for the Prym M-theta function, we take

$$\begin{aligned} \xi _j=\mathop {\textstyle \sum }\limits _{k=\text {odd}}q_j^kt_k=\left\{ \begin{array}{lll} \mathop {\textstyle \sum }\limits _{k=\text {odd}}b_n^kt_k,&{}\quad \text {if}~j=2n-1,\\ \mathop {\textstyle \sum }\limits _{k=\text {odd}}c_n^kt_k,&{}\quad \text {if}~j=2n. \end{array}\right. \quad \text {for}\quad 1\le j\le 2g. \end{aligned}$$

(The time variables can include $t_{-1}$ [12].) Then, the element of the skew symmetric matrix Q in (2.9), ${\tilde{\vartheta }}^B_g(z;{\tilde{\Omega }}^B)=\text {Pf}(Q)$ can be expressed by

$$\begin{aligned} Q_{j,k}=c_{j,k}+\int ^x D_1 e^{\xi _j}\cdot e^{\xi _k}\,dx, \end{aligned}$$

where $D_1$ is the Hirota derivative with respect to $t_1$, and the $\tau$-function (2.9) gives a solution of the BKP equation [12],

$$\begin{aligned} (D_1^6-5D_1^3D_3-5D_3^2+9D_1D_5)\tau \cdot \tau =0. \end{aligned}$$

In a future communication, we plan to prove a classification theorem for the regular BKP solitons similar to Theorem 3.10.

4 Examples

4.1 Hirota type soliton solutions

First note that the M-theta function (2.7) gives the Hirota type g-soliton solution [12] by choosing the following variables with $\{b_j,c_j\}=\pi ^{-1}(p_j)$,

$$\begin{aligned} {\tilde{\phi }}_j=(c_j-b_j)x+(c_j^2-b_j^2)y+(c_j^3-b_j^3)t+\phi _j^0,\quad j=1,\ldots ,g, \end{aligned}$$

where $b_j,c_j$ and $\phi _j^0$ are arbitrary constants. The coefficients $C_{k,l}:=e^{2\pi i{\tilde{\Omega }}_{k,l}}$ are

$$\begin{aligned} C_{k,l}=\frac{(b_k-b_l)(c_k-c_l)}{(b_k-c_l)(c_k-b_l)}. \end{aligned}$$

For the regular real solitons, we assume all the parameters $\{b_j, c_j, \phi _j^0\}$ be real with $b_j<c_j$ and $C_{k,l}>0$, which leads to the following conditions,

• (a) $b_k<b_l<c_l<c_k$, or

• (b) $b_k<c_k<b_l<c_l$,

(the case (a) is called P-type, and the case (b) is called O-type in [16]). In terms of the matrix $A\in \text {Gr}(g,2\,g)$, the $b_j$ corresponds to a pivot position ($b_1<\cdots <b_g$) and the $c_j$ to a non-pivot position. For example, $A_{(a)}$ for the case (a) and $A_{(b)}$ for the case (b) in $\text {Gr}(2,4)$ are given by

$$\begin{aligned} A_{(a)}=\begin{pmatrix} 1 &{}0 &{} 0 &{} -a\\ 0&{}1&{} b&{} 0 \end{pmatrix}\quad \text {and}\quad A_{(b)}=\begin{pmatrix} 1&{}a&{}0&{}0\\ 0&{}0&{}1&{}b \end{pmatrix}, \end{aligned}$$

that is, for $A_{(a)}$, $b_1=\kappa _1, c_1=\kappa _4, b_2=\kappa _2, c_2=\kappa _3$, and for $A_{(b)}$, $b_1=\kappa _1, c_1=\kappa _2, b_2=\kappa _3, c_2=\kappa _4$. In general, the matrix A consists of both (a) and (b) types. For example, we have

$$\begin{aligned} A=\begin{pmatrix} 1&{}0&{}0&{}0&{}0&{}-a\\ 0&{}1&{}b&{}0&{}0&{}0\\ 0&{}0&{}0&{}1&{}c&{}0 \end{pmatrix}. \end{aligned}$$

The $\tau$-function in the form of the M-theta function is given as follows. First note that $|{\mathcal {M}}_1(A)|=3$, i.e., there are 3 singular points, $S=\{p_1,p_2,p_3\}$, and we have

$$\begin{aligned} \phi _1=[1,6]=\xi _6-\xi _1,\quad \phi _2=[2,3]=\xi _3-\xi _2,\quad \phi _3=[4,5]=\xi _5-\xi _4, \end{aligned}$$

which gives $\pi ^{-1}(p_j)=\{b_j,c_j\}$, the inverse image of the normalization,

$$\begin{aligned} b_1=\kappa _1,\quad c_1=\kappa _6,\qquad b_2=\kappa _2,\quad c_2=\kappa _3,\qquad b_3=\kappa _4,\quad c_3=\kappa _5. \end{aligned}$$

For the free parameters $\{a,b,c\}$ in the matrix A, we introduce $\phi ^0_j$ as

$$\begin{aligned} e^{\phi _1^0}&=a\frac{(\kappa _2-\kappa _4)(\kappa _2-\kappa _6)(\kappa _4-\kappa _6)}{(\kappa _1-\kappa _2)(\kappa _1-\kappa _4)(\kappa _2-\kappa _4)},\\ e^{\phi _2^0}&=b\frac{(\kappa _1-\kappa _3)(\kappa _1-\kappa _4)(\kappa _3-\kappa _4)}{(\kappa _1-\kappa _2)(\kappa _1-\kappa _4)(\kappa _2-\kappa _4)},\\ e^{\phi _3^0}&=c\frac{(\kappa _1-\kappa _2)(\kappa _1-\kappa _5)(\kappa _2-\kappa _5)}{(\kappa _1-\kappa _2)(\kappa _1-\kappa _4)(\kappa _2-\kappa _4)}, \end{aligned}$$

Then, the M-theta function is given by

$$\begin{aligned} {\tilde{\vartheta }}_3^{(3)}&=1+\mathop {\textstyle \sum }\limits _{j=1}^3e^{{\tilde{\phi }}_j}+\mathop {\textstyle \sum }\limits _{1\le j<k\le 3}C_{j,k}e^{{\tilde{\phi }}_j+{\tilde{\phi }}_k}+C_{1,2}C_{1,3}C_{2,3}e^{{\tilde{\phi }}_1+{\tilde{\phi }}_2+{\tilde{\phi }}_3}, \end{aligned}$$

where ${\tilde{\phi }}_j=\phi _j+\phi _j^0$ with $C_{k,l}$ given by (2.3).

In general, the soliton solution of Hirota type consists of g line solitons, and if $b_k+c_k\ne b_l+c_l$, the corresponding solitons $\phi _k=[b_k,c_k]$ and $\phi _l=[b_l,c_l]$ intersect with phase shift computed by $C_{k,l}$. Depending of the types (a) or (b) above, we have $C_{k,l}<1$ (negative phase shift) for (a), and $C_{k,l}>1$ (positive phase shift) for (b).

Remark 4.1

We note that the dual graph of the soliton graph (defined in [16]) gives a rhombus tiling with $\left( {\begin{array}{c}g\\ 2\end{array}}\right)$ rhombuses, in which each rhombus is labeled by the intersecting two solitons $\phi _k$ and $\phi _l$. Then, coloring each rhombus according to the types either $C_{k,l}>1$ or $C_{k,l}<1$, we have a colored rhombus tiling of 2g-gon. The total number of different colored tilings is given by a Catalan number [4, 27],

$$\begin{aligned} C_g=\frac{1}{g+1}\left( {\begin{array}{c}2g\\ g\end{array}}\right) . \end{aligned}$$

It is then interesting to notice that a KP soliton with a set of random $C_{k,l}$’s (corresponding to the positive and negative phase shifts) may provide a two-dimensional extension of the KdV soliton gas (see [28] and the references therein).

4.2 A degenerate example

Consider an $2\times 4$ matrix A in the positive $\text {Gr}(2,4)$, which is given by

$$\begin{aligned} A=\begin{pmatrix} 1 &{} 0 &{} -a &{} -b\\ 0&{} 1 &{} c &{} d \end{pmatrix} \end{aligned}$$

where a, b, c, d are positive constants with $cb-ad>0$. We introduce the new variables,

$$\begin{aligned} \phi _1:=[1,3]= & {} \xi _3-\xi _1,\quad \phi _2:=[1,4]=\xi _4-\xi _1,\quad \\ \phi _3:=[2,3]= & {} \xi _3-\xi _2,\quad \phi _4:=[2,4]=\xi _4-\xi _2. \end{aligned}$$

This implies that the pairs of points $\pi ^{-1}(p_j)=\{b_j,c_j\}$ of the normalization for ${{\tilde{g}}}=4$ are given by

$$\begin{aligned} \begin{matrix} b_1=\kappa _1,\quad &{} b_2=\kappa _1,\quad &{} b_3=\kappa _2,\quad &{} b_4=\kappa _2,\\ c_1=\kappa _3, &{} c_2=\kappa _4, &{}\kappa _3=\kappa _3, &{} c_4=\kappa _4. \end{matrix} \end{aligned}$$

(4.1)

Note here that some of the points are also identified, and it leads to further degenerations of $\Omega _{k,l}\rightarrow +i\infty$ (see below). Also note that these phases are linearly dependent, that is,

$$\begin{aligned} \phi _4=[2,4]=[2,3]+[3,1]+[1,4]=\phi _3+\phi _2-\phi _1, \end{aligned}$$

which implies that $\text {dim}\,P_1(A)=3$. We now introduce the phase constants $e^{\phi _j^0}$ to represent the nonzero parameters in A except the pivot ones,

$$\begin{aligned} e^{\phi _1^0}:=&a\frac{\kappa _3-\kappa _2}{\kappa _2-\kappa _1},\quad e^{\phi _2^0}:=b\frac{\kappa _4-\kappa _2}{\kappa _2-\kappa _1},\quad e^{\phi _3^0}:=c\frac{\kappa _3-\kappa _1}{\kappa _2-\kappa _1},\quad e^{\phi _4^0}:=d\frac{\kappa _4-\kappa _1}{\kappa _2-\kappa _1}. \end{aligned}$$

The phase variables are then given by $2\pi iz_k={\tilde{\phi }}_k=\phi _k+\phi _k^0,\quad \text {for}~\, k=1,\ldots , 4$, and the coefficients $C_{k,l}:=e^{2\pi i{\tilde{\Omega }}_{k,l}}=\frac{(b_k-b_l)(c_k-c_l)}{(b_k-c_l)(c_k-b_l)}$ become (4.1),

$$\begin{aligned}{} & {} C_{1,2}=C_{1,3}=C_{2,4}=C_{3,4}=0,\quad C_{2,3}=\frac{(\kappa _1-\kappa _2)(\kappa _3-\kappa _4)}{(\kappa _1-\kappa _3)(\kappa _2-\kappa _4)},\\{} & {} C_{1,4}=-\frac{(\kappa _1-\kappa _2)(\kappa _3-\kappa _4)}{(\kappa _1-\kappa _4)(\kappa _2-\kappa _3)}. \end{aligned}$$

Then, the $\tau$-function for the matrix A can be expressed in the form of ${\tilde{\vartheta }}_4^{(4)}$, i.e., $\tau _A=E_{1,2}{\tilde{\vartheta }}_4^{(4)}$ with

$$\begin{aligned} {\tilde{\vartheta }}_4^{(4)}&=1+e^{{\tilde{\phi }}_1}+e^{{\tilde{\phi }}_2}+e^{{\tilde{\phi }}_3}+e^{{\tilde{\phi }}_4}\nonumber \\&\quad +\frac{(\kappa _1-\kappa _2)(\kappa _3-\kappa _4)}{(\kappa _1-\kappa _3)(\kappa _2-\kappa _4)}e^{{\tilde{\phi }}_2+{\tilde{\phi }}_3} -\frac{(\kappa _1-\kappa _2)(\kappa _3-\kappa _4)}{(\kappa _1-\kappa _4)(\kappa _2-\kappa _3)}e^{{\tilde{\phi }}_1+{\tilde{\phi }}_4}. \end{aligned}$$

(4.2)

where $E_{1,2}=(\kappa _2-\kappa _1)e^{\xi _1+\xi _2}$. Here, the additional degenerations in $\Omega _{k,l}$ are

$$\begin{aligned} {\tilde{\Omega }}_{1,2}={\tilde{\Omega }}_{1,3}={\tilde{\Omega }}_{2,4}={\tilde{\Omega }}_{3,4}=+i\infty . \end{aligned}$$

This implies that the interactions between these waves give an infinite phase shift, which represents a resonant interaction. Note here that $C_{1.4}=e^{2\pi i{\tilde{\Omega }}_{1,4}}<0$, i.e., the real part of ${\tilde{\Omega }}_{1,4}$ is $\text {Re}({\tilde{\Omega }}_{1,4})=\frac{1}{2}$. Then, the positivity of $\Delta _{3,4}(A)=cb-ad$ is due to the values of $\phi ^0_1+\phi _4^0$ and $\phi ^0_2+\phi ^0_3$.

Remark 4.2

The dimension of $P_1(A)$ represents of the number of independent phases. In this example of $\text {Gr}(2,4)$, the dimension is three, but the KP soliton should be embedded into ${{\tilde{g}}}=4$ M-theta function. This implies that the set of singular points on the curve is $S=\{p_1,\ldots ,p_4\}$. We will discuss the details of the singular structures of the curve associated to KP solitons in a future communication.

Remark 4.3

As is also mentioned in [1], the KP solitons can be considered as tropical limits of the theta function, and the tropical Riemann period matrix defines a Delaunay polytope (dual graph is the soliton graph [17]). The dimension of the polytope should be considered as dim$P_1(A)$, not ${{\tilde{g}}}=|{\mathcal {M}}_1(A)|$. For the example above of $\text {Gr}(2,4)$, we have $\text {dim}P_1(A)=3$, and the corresponding polytope is octahedron. In [1], it is shown that there are total of 5 distinct polytopes such as tetrahedra, octahedra, square pyramid, triangular prism and cube. They are easily identified as the (dual) soliton graphs in (x, y, t) space. The higher dimensional examples are obtained by the KP hierarchy (see, e.g., [20]). We will discuss the details in a future communication.

5 KP solitons on a quasi-periodic background

In the limit (2.2), if we consider only for some $\Omega _{j,j}\rightarrow +i\infty$, we get solitons on a quasi-periodic background. Let us consider the limit with $\Omega _{i,i}\rightarrow +i\infty$ for $1\le i\le n<g$. Then, having the shift $z_i\rightarrow z_i-\frac{1}{2}\Omega _{i.i}$ for $1\le i\le n$, we have

$$\begin{aligned} \vartheta _g({\textbf{z}};\Omega )~\rightarrow ~{\tilde{\vartheta }}^{(n)}_g({\textbf{z}};{\tilde{\Omega }})&:= \vartheta _{g-n}({\textbf{z}}^{(n)};{\tilde{\Omega }}^{(n)})+ \mathop {\textstyle \sum }\limits _{j=1}^ne^{2\pi iz_j}\vartheta _{g-n}({\textbf{z}}^{(n)}+{\tilde{\Omega }}_j\,;\,{\tilde{\Omega }}^{(n)})\nonumber \\&\quad +\mathop {\textstyle \sum }\limits _{1\le j\le k\le n}e^{2\pi i({\tilde{\Omega }}_{j,k}+z_j+z_k)}\vartheta _{g-n}({\textbf{z}}^{(n)}+{\tilde{\Omega }}_j+{\tilde{\Omega }}_k\,;\,{\tilde{\Omega }}^{(n)})\nonumber \\&\quad +\cdots +e^{2\pi i(\mathop {\textstyle \sum }\limits _{1\le j<k\le n}{\tilde{\Omega }}_{j,k}+\mathop {\textstyle \sum }\limits _{j=1}^nz_j)}\vartheta _{g-n}({\textbf{z}}^{(n)}+\mathop {\textstyle \sum }\limits _{j=1}^n{\tilde{\Omega }}_j\,;\,{\tilde{\Omega }}^{(n)}), \end{aligned}$$

(5.1)

where we have defined

$$\begin{aligned} {\textbf{z}}^{(n)}=(z_{n+1},z_{n+2},\cdots ,z_g),\qquad {\textbf{z}}^{(n)}+{\tilde{\Omega }}_j=(z_{n+1}+{\tilde{\Omega }}_{j,n+1},\ldots , z_g+{\tilde{\Omega }}_{j,g}). \end{aligned}$$

and ${\tilde{\Omega }}^{(n)}$ is the period matrix given by the bottom $(g-n)\times (g-n)$ part of the period matrix ${\tilde{\Omega }}$, i.e., ${\tilde{\Omega }}^{(n)}=({\tilde{\Omega }}_{k,l})_{n+1\le k,l\le g}$. The $\vartheta _{g-n}({\textbf{z}}^{(n)};{\tilde{\Omega }}^{(n)})$ is the theta function of genus $g-n$ with the period matrix ${\tilde{\Omega }}^{(n)}$. One can express the function ${\tilde{\vartheta }}^{(n)}_g$ using the vertex operator (2.6),

$$\begin{aligned} {\tilde{\vartheta }}^{(n)}_g({\textbf{z}};{\tilde{\Omega }})=\prod _{j=1}^n(1+{{\hat{V}}}_j[{\tilde{\Omega }}])\cdot \vartheta _{g-n}({\textbf{z}}^{(n)};{\tilde{\Omega }}^{(n)}). \end{aligned}$$

(5.2)

(Similar formula has been presented in Nakayashiki’s lecture from different point of view [25].) From Lemma 2.1, we see that the coefficients of the $\vartheta _{g-n}({\textbf{z}}^{(n)};{\tilde{\Omega }}^{(n)})$ depends only on the variables $z_j$ for $1\le j\le n$ and the matrix ${\tilde{\Omega }}_{i,j}$ for $1\le i<j\le n$. Then, considering a matrix $A\in \text {Gr}(N,M)$ with $|{\mathcal {M}}_1(A)|=n$, one can construct ${\tilde{\Omega }}=({\tilde{\Omega }}_{i,j})_{1\le i,j\le n}$ for a KP soliton following Theorem 3.10. We then expect to have the KP soliton on quasi-periodic background given by the theta function $\vartheta _{g-n}(z^{(n)};{\tilde{\Omega }})$.

The wave pattern of the KP soliton has a web-like structure, which divides several connected regions of the plane (see, e.g., [16]). Each connected region represents the dominant exponential in the $\tau$-function. Then, the solution generated by (5.2) is expected to have the similar structure of the KP soliton where each connected region presents a part of quasi-periodic solution generated by $\vartheta _{g-n}$. And each soliton provides the phase shift to the adjacent quasi-periodic solutions. For example, consider the case with $g=2$ and $n=1$, and take the limit $\Omega _{1,1}\rightarrow +i\infty$. We have

$$\begin{aligned} \vartheta _2^{(1)}(z_1,z_2,{\tilde{\Omega }})=(1+{{\hat{V}}}_1)\vartheta _1(z_2;{\tilde{\Omega }}_{2,2})=\vartheta _1(z_2;{\tilde{\Omega }}_{2,2})+e^{2\pi iz_1}\vartheta _1(z_2+{\tilde{\Omega }}_{1,2};{\tilde{\Omega }}_{2,2}). \end{aligned}$$

This implies that in the region where $2\pi i z_1=\phi _1\ll 0$, we have the periodic solution generated by $\tau (x,y,t)=\vartheta _1(ux+vy+wt;{\tilde{\Omega }}_{2,2})$ (i.e., we take $z_2=ux+vy+wt$), and in the region where $2\pi iz_1={\tilde{\phi }}_1\gg 0$, we have the same periodic solution with the phase shift given by $2\pi i{\tilde{\Omega }}_{1,2}$. That is, one KP soliton interacts with the periodic background with phase shift. In his study of degeneration of hyperelliptic curve [24], Nakayashiki has obtained a similar formula [26]. Also in [3, 13], the authors obtained a similar result for the KdV case. We will discuss the general structure of the KP solitons on quasi-periodic background in a future communication.