On non-autonomous differential-difference AKP, BKP and CKP equations

Abstract

Based on the direct linearization framework of the discrete Kadomtsev–Petviashvili-type equations presented in the work of Fu & Nijhoff (Fu W, Nijhoff FW. 2017 Direct linearizing transform for three-dimensional discrete integrable systems: the lattice AKP, BKP and CKP equations. Proc. R. Soc. A 473, 20160915 (doi:10.1098/rspa.2016.0915)), six novel non-autonomous differential-difference equations are established, including three in the AKP class, two in the BKP class and one in the CKP class. In particular, one in the BKP class and the one in the CKP class are both in (2 + 2)-dimensional form. All the six models are integrable in the sense of having the same linear integral equation representations as those of their associated discrete Kadomtsev–Petviashvili-type equations, which guarantees the existence of soliton-type solutions and the multi-dimensional consistency of these new equations from the viewpoint of the direct linearization.

1. Introduction

The study of discrete integrable systems has been becoming one of the most prominent branches in the theory of integrable systems in the past two decades, resulting in the establishment of many novel concepts and theories in modern mathematics, see e.g. [1]. There are several methods of constructing integrable discretization of nonlinear differential equations. Among those, a very effective one is to construct the Bäcklund transform (BT) and the superposition formula of a partial differential equation (PDE), and consider them as the associated differential-difference equation (DΔE) and partial difference equation (PΔE), respectively (see [2,3]). Compared with integrable PDEs, the associated integrable PΔEs seem to possess a much richer structure, which reflects the fact that the discrete equations encode the information of the whole hierarchy of the corresponding continuous equations in an implicit way.

Among the theory of discrete integrable systems, there are three important scalar PΔEs in three dimensions. The first one is the discrete Kadomtsev–Petviashvili (KP) equation (often known as the Hirota equation or the Hirota–Miwa (HM) equation)

$$\begin{array}{r}(p_i-p_j)({\mathrm{T}}_{n_h}\tau )({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}\tau )+(p_j-p_h)({\mathrm{T}}_{n_i}\tau )({\mathrm{T}}_{n_j}{\mathrm{T}}_{n_h}\tau )+(p_h-p_i)({\mathrm{T}}_{n_j}\tau )({\mathrm{T}}_{n_h}{\mathrm{T}}_{n_i}\tau )=0,\end{array}$$ 1.1

in which the dependent variable τ is a function of the discrete arguments n_j and the lattice parameters p_j for j = 1, 2, …, the notation ${\mathrm{T}}_{n_j}$ stands for the forward shift operator with respect to the corresponding discrete argument n_j, and p_h, p_i and p_j are distinct. This equation was introduced by Hirota [4] as a discrete analogue of the generalized Toda equation, and here we adopt the form in (1.1) which possesses explicit discrete soliton solutions. Since equation (1.1) is associated with the infinite-dimensional algebra A_∞, here we also refer to it as the discrete AKP equation, in order to distinguish it from the other two discrete KP-type equations below. The second one is the discrete BKP equation (also referred to as the Miwa equation)

$$\begin{array}{rl}\hspace{0.17em}& (p_h-p_i)(p_i-p_j)(p_j-p_h)\tau ({\mathrm{T}}_{n_h}{\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}\tau )+(p_h+p_i)(p_h+p_j)(p_i-p_j)({\mathrm{T}}_{n_h}\tau )({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}\tau )\\ \hspace{0.17em}& +(p_i+p_j)(p_i+p_h)(p_j-p_h)({\mathrm{T}}_{n_i}\tau )({\mathrm{T}}_{n_h}{\mathrm{T}}_{n_j}\tau )+(p_j+p_h)(p_j+p_i)(p_h-p_i)({\mathrm{T}}_{n_j}\tau )({\mathrm{T}}_{n_h}{\mathrm{T}}_{n_i}\tau )=0,\end{array}$$ 1.2

which was introduced by Miwa in [5]. Although (1.2) has an additional term in comparison with (1.1), it is actually a special reduction of the discrete AKP equation (algebraically its associated algebra B_∞ is a sub-algebra of A_∞). The third model is the discrete CKP equation (also called the hyperdeterminant equation)

$${(A_1+A_2-A_3-A_4)}^2=4B_1{B}_2,$$ 1.3

where the expressions A_i and B_i are given by

$$\begin{array}{rl}\hspace{0.17em}& A_1={(p_h-p_i)}^2{(p_i-p_j)}^2{(p_j-p_h)}^2\tau ({\mathrm{T}}_{n_h}{\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}\tau ),\\ \hspace{0.17em}& A_2={(p_h+p_i)}^2{(p_h+p_j)}^2{(p_i-p_j)}^2({\mathrm{T}}_{n_h}\tau )({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}\tau ),\\ \hspace{0.17em}& A_3={(p_i+p_j)}^2{(p_i+p_h)}^2{(p_j-p_h)}^2({\mathrm{T}}_{n_i}\tau )({\mathrm{T}}_{n_h}{\mathrm{T}}_{n_j}\tau ),\\ \hspace{0.17em}& A_4={(p_j+p_h)}^2{(p_j+p_i)}^2{(p_h-p_i)}^2({\mathrm{T}}_{n_j}\tau )({\mathrm{T}}_{n_h}{\mathrm{T}}_{n_i}\tau ),\\ \hspace{0.17em}& B_1=(p_h^2-p_i^2)(p_j^2-p_h^2)[{(p_i+p_j)}^2({\mathrm{T}}_{n_i}\tau )({\mathrm{T}}_{n_j}\tau )-{(p_i-p_j)}^2\tau ({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}\tau )],\\ \text{and}& B_2={\mathrm{T}}_{n_h}{B}_1,\end{array}$$

respectively, which is also a reduced equation from the discrete AKP equation. equation (1.3) appeared in Kashaev’s paper [6] for the first time from the star-triangle transform in the Ising model, and it was later identified by Schief [7] that such a model actually describes the superposition formula for the continuous KP equation of C_∞-type. The form in (1.3) that contains lattice parameters in the coefficients was given by the authors in [8] in order to construct its soliton-type solution.

The discrete KP-type equations are not only remarkable in their own right in the integrable systems theory, since these equations, as higher-dimensional models, can reduce to many lower-dimensional integrable discrete systems, such as the famous discrete Korteweg–de Vries (KdV) and Boussinesq (BSQ) systems, see e.g. [9,10], but also play crucial roles in other subjects in modern mathematics, especially in geometry. It was shown by Konopelchenko and Schief that the discrete AKP, BKP and CKP equations are connected to fundamental theorems of plane geometry, i.e. Menelaus’ theorem [11], reciprocal quadrangles [12] and Carnot’s theorem [7], respectively. Besides, Doliwa pointed out that these models also arise in the multi-dimensional consistency (MDC) quadrilateral lattice theory in discrete geometry [13–15]. Very recently, it was also revealed that the discrete KP-type equations are closely related to discrete line complexes [16] and circle complexes [17].

In addition to PΔEs, there are also the so-called generating PDEs, where the terminology ‘generating’ follows from the fact that it generates a whole continuous integrable hierarchy. From the perspective of integrable systems, the key step of constructing generating PDEs is to take the lattice parameters in the associated PΔEs as independent variables. This technique is actually a reflection of introducing Miwa’s coordinates, see [18]. To be more precise, the derivative with respect to a lattice parameter is equivalent to the higher-order derivatives in terms of the continuous flow variables, resulting in the higher-order symmetries in an integrable hierarchy. The first example was proposed in [19] for the KdV class in the form of a non-autonomous fourth-order nonlinear equation, and it is integrable in the sense that it possesses soliton, Lax pair, Lagrangian structure, and Painlevé reduction, etc. More importantly, it was shown in [20] that a proper generalization of this equation incorporates the hyperbolic Ernst equation for a Weyl neutrino field in general relativity. In the spirit of this, a class of generating PDEs for the BSQ family, also as integrable nonlinear models, were constructed in [21], and they represent the hyperbolic Ernst equations for a source-free Maxwell field and a Weyl neutrino field, as a generalization for the previous result of KdV.

It turns out that it is very difficult to derive closed-form equations for a scalar field that would be the analogues of the generating PDEs for the KP-type hierarchies. However, as we will show in this paper, it is possible to derive semi-discrete analogues (in the form of DΔE) of these generating PDEs for the KP-type equations, which simultaneously contain lattice variables and lattice parameters as independent variables. Such DΔEs, as non-autonomous semi-discrete equations, are significant in integrable systems theory. In fact, on the two-dimensional level, these semi-discrete equations appear in similarity reductions to discrete Painlevé equations [22], and also play roles of master symmetries of two-dimensional integrable difference equations, see e.g. [23,24].

We aim to construct the associated non-autonomous DΔEs for the KP-type equations, based on the direct linearization (DL) framework for the KP-type equations given in [8]. The DL method was proposed by Fokas and Ablowitz (e.g. [25,26]) to solve the initial value problems for the KdV and KP equations, as a generalization of the well-known Riemann–Hilbert problem [27]. Subsequently, it was developed into a powerful tool to systematically study integrable structures behind families of discrete and continuous nonlinear equations and their interrelations, e.g. [3,10,22,28–30]. The key idea in the DL is to associate a nonlinear equation with a linear integral equation, and by introducing the infinite matrix structure, it allows us to observe the integrability and solution structures of a nonlinear system simultaneously. Recently, the link between the linear integral equation and several affine Lie algebras was further established. With the help of considering reductions on the measure of the linear integral equation, the DL scheme for the discrete AKP, BKP and CKP equations was proposed [8]. This makes it possible to further study the associated non-autonomous DΔEs for these KP-type equations from the general framework.

In this paper, we establish six non-autonomous DΔEs. They are

$$\begin{array}{rl}\hspace{0.17em}& \frac{1}{2}{(p_i-p_j)}^2{\mathrm{D}}_{p_i}{\mathrm{D}}_{p_j}\tau \cdot \tau =n_i{n}_j[({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}^{-1}\tau )({\mathrm{T}}_{n_i}^{-1}{\mathrm{T}}_{n_j}\tau )-{\tau }^2],\end{array}$$ 1.4a

$$\begin{array}{rl}\hspace{0.17em}& (p_i-p_j){\mathrm{D}}_{p_j}({\mathrm{T}}_{n_i}\tau )\cdot \tau =n_j[\tau ({\mathrm{T}}_{n_i}\tau )-({\mathrm{T}}_{n_j}\tau )({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}^{-1}\tau )],\\ \text{and}& (p_i-p_j){\mathrm{D}}_{p_j}[\tau ({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}\tau )]\cdot [({\mathrm{T}}_{n_i}\tau )({\mathrm{T}}_{n_j}\tau )]\end{array}$$ 1.4b

$$\begin{array}{rl}\hspace{0.17em}& =n_j{({\mathrm{T}}_{n_j}\tau )}^2({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}\tau )({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}^{-1}\tau )-(n_j+1)\tau {({\mathrm{T}}_{n_i}\tau )}^2({\mathrm{T}}_{n_j}^2\tau )+\tau ({\mathrm{T}}_{n_i}\tau )({\mathrm{T}}_{n_j}\tau )({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}\tau )\end{array}$$ 1.4c

in the AKP class,¹

$$\begin{array}{rl}2p_i{p}_j(p_i^2-p_j^2){\mathrm{D}}_{p_i}{\mathrm{D}}_{p_j}\tau \cdot \tau & =n_i{n}_j[{(p_i+q_j)}^4({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}^{-1}\tau )({\mathrm{T}}_{n_i}^{-1}{\mathrm{T}}_{n_j}\tau )\\ \hspace{0.17em}& -{(p_i-q_j)}^4({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}\tau )({\mathrm{T}}_{n_i}^{-1}{\mathrm{T}}_{n_j}^{-1}\tau )-8p_i{p}_j(p_i^2+p_j^2){\tau }^2]\end{array}$$ 1.5a

and

$$\begin{array}{rl}\hspace{0.17em}& 2p_j(p_i^2-p_j^2){\mathrm{D}}_{p_j}({\mathrm{T}}_{n_i}\tau )\cdot \tau =n_j[4p_i{p}_j\tau ({\mathrm{T}}_{n_i}\tau )+{(p_i-p_j)}^2({\mathrm{T}}_{n_j}^{-1}\tau )({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}\tau )-{(p_i+p_j)}^2({\mathrm{T}}_{n_j}\tau )({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}^{-1}\tau )]\end{array}$$ 1.5b

in the BKP class, and

$$\begin{array}{rl}\hspace{0.17em}& p_i{p}_j{(p_i^2-p_j^2)}^2{\mathrm{D}}_{p_i}{\mathrm{D}}_{p_j}\tau \cdot \tau =n_i{n}_j[{(p_i-p_j)}^2{E}^{1/2}{F}^{1/2}+{(p_i+p_j)}^2{G}^{1/2}{H}^{1/2}-8p_i{p}_j(p_i^2+p_j^2){\tau }^2]\end{array}$$ 1.6

in the CKP class, where E, F, G and H are given by

$$\begin{array}{rl}\hspace{0.17em}& E={(p_i+p_j)}^2({\mathrm{T}}_{n_i}^{-1}\tau )({\mathrm{T}}_{n_j}^{-1}\tau )-{(p_i-p_j)}^2\tau ({\mathrm{T}}_{n_i}^{-1}{\mathrm{T}}_{n_j}^{-1}\tau ),\\ \hspace{0.17em}& F={(p_i+p_j)}^2({\mathrm{T}}_{n_i}\tau )({\mathrm{T}}_{n_j}\tau )-{(p_i-p_j)}^2\tau ({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}\tau ),\\ \hspace{0.17em}& G={(p_i-p_j)}^2({\mathrm{T}}_{n_i}^{-1}\tau )({\mathrm{T}}_{n_j}\tau )-{(p_i+p_j)}^2\tau ({\mathrm{T}}_{n_i}^{-1}{\mathrm{T}}_{n_j}\tau )\\ \text{and}& H={(p_i-p_j)}^2({\mathrm{T}}_{n_i}\tau )({\mathrm{T}}_{n_j}^{-1}\tau )-{(p_i+p_j)}^2\tau ({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}^{-1}\tau ),\end{array}$$

respectively, where the notation D_· stands for Hirota’s bilinear derivative (e.g. [31]) with respect to the corresponding arguments p_i and p_j, which is defined by

$${\mathrm{D}}_{x}f\cdot g=({\mathrm{\partial }}_x-{\mathrm{\partial }}_{x^{\prime }})f(x)g(x^{\prime }){|}_{x^{\prime }=x},$$

for arbitrary differentiable functions f(x) and g(x).

The tau functions for the above equations have their precise definitions in terms of infinite matrices and double integral with regard to the spectral variables, which will be given in the sections of the derivations of these equations. We note that an autonomous version of (1.5a) was given in [32] very recently, which plays the role of a higher-order semi-discrete BKP equation (i.e. a symmetry) and was referred to as the (2+2)-dimensional Toda lattice. But here equation (1.5a) is a non-autonomous equation having the lattice parameters as the continuous independent variables, potentially acting as the master symmetry of the discrete BKP equation.

The paper is organized as follows. Section 2 concerns the formal structure of the direct linearization approach. In §§3, 4 and 5, we provide derivations of (1.4a), (1.4b) and (1.4c) in the AKP class, (1.5a) and (1.5b) in the BKP class, as well as (1.6) in the CKP class, respectively. The brief discussion on the integrability of the six DΔEs is made in §6.

2. Formal structure of the direct linearization

In [8], the discrete AKP, BKP and CKP equations were studied within a single framework, given by the direct linearizing transform (DLT), which provides a dressing-type scheme for obtaining new solutions from given seed solutions for those integrable equations. By contrast, what we mean by DL is a special case where the seed corresponds to a ‘free’ solution (namely when the initial solution is trivial) in the direct linearising transform (DLT). The latter restriction is useful if we want to derive new equations from some basic assumptions about the initial solutions.

We start with introducing some fundamental infinite matrices and vectors and their properties which are needed in the DL framework. First, we need a rank 1 projection matrix

$$\mathit{O}=\left(\begin{array}{ccccc}\ddots & \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& 0& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& \boxed{1}& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& 0& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& \ddots \end{array}\right),$$ 2.1

where the ‘box’ denotes the location of the central element, namely the (0, 0)-entry of the matrix. This infinite matrix has the property

$${(\mathit{O}\mathit{U})}_{i,j}={\delta }_{i,0}{U}_{0,j}\text{and}{(\mathit{U}\mathit{O})}_{i,j}=U_{i,0}{\delta }_{0,j},\text{with}\hspace{0.17em}{\delta }_{i,j}=\{\begin{array}{ll}1,& i=j,\\ 0,& i\ne j,\end{array}\mathrm{\forall }\hspace{0.17em}i,j\in \mathbb{Z},$$

for an arbitrary infinite matrix

$$\mathit{U}=\left(\begin{array}{ccccc}\hspace{0.17em}& ⋮& ⋮& ⋮& \hspace{0.17em}\\ \cdots & U_{-1,-1}& U_{-1,0}& U_{-1,1}& \cdots \\ \cdots & U_{0,-1}& \boxed{U_{0,0}}& U_{0,1}& \cdots \\ \cdots & U_{1,-1}& U_{1,0}& U_{1,1}& \cdots \\ \hspace{0.17em}& ⋮& ⋮& ⋮& \hspace{0.17em}\end{array}\right),$$

where the operation ( · )_i,j denotes taking the (i, j)-entry.

Next, we introduce two infinite matrices

$$\mathbf{\Lambda }=\left(\begin{array}{ccccc}\ddots & \ddots & \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& 0& 1& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& \boxed{0}& 1& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& 0& \ddots \\ \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& \ddots \end{array}\right)\text{and}{\hspace{0.17em}}^t\mathbf{\Lambda }=\left(\begin{array}{ccccc}\ddots & \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}\\ \ddots & 0& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& 1& \boxed{0}& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& 1& 0& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& \ddots & \ddots \end{array}\right).$$ 2.2

The matrices Λ and ^tΛ are the transpose of each other, and have properties

$${(\mathbf{\Lambda }\hspace{0.17em}\mathit{U})}_{i,j}=U_{i+1,j}\text{and}{(\mathit{U}\hspace{0.17em}{\hspace{0.17em}}^t\mathbf{\Lambda })}_{i,j}=U_{i,j+1},$$

namely the multiplications by Λ and ^tΛ raise the row index and the column index of U, respectively. For this reason, we refer to them as index-raising matrices. We also introduce the infinite-dimensional vectors

$${\mathit{c}}_k={(\cdots ,k^{-1},1,k,\hspace{0.17em}\dots )}^{\mathrm{T}}\text{and}{\hspace{0.17em}}^t{\mathit{c}}_{k^{\prime }}=(\cdots ,k^{\prime -1},1,k^{\prime },\hspace{0.17em}\dots ),$$ 2.3

which obey the following identities:

$$\mathbf{\Lambda }\hspace{0.17em}{\mathit{c}}_k=k\hspace{0.17em}{\mathit{c}}_k,{\hspace{0.17em}}^t{\mathit{c}}_{k^{\prime }}{\hspace{0.17em}}^t\mathbf{\Lambda }=k^{\prime }{\hspace{0.17em}}^t{\mathit{c}}_{k^{\prime }}.$$

In the sections below, we also need the notions of trace and determinant for an infinite matrix. We give the formal definition of a trace of an arbitrary infinite matrix U as follows:

$$\text{tr}\hspace{0.17em}\mathit{U}=\sum _{i\in \mathbb{Z}}{U}_{i,i}.$$ 2.4

Note that this is a formal definition, since the infinite summation with respect to i over the integer ring may lead to divergence. In order to avoid this issue, in this paper we only deal with the trace of a matrix involving the projection matrix O, in which case the trace is always convergent. For example, we have

$$\text{tr}(\mathit{O}\mathit{U})=\text{tr}(\mathit{U}\mathit{O})=U_{0,0}={(\mathit{U})}_{0,0}.$$

In the convergent case, the trace also satisfies tr(UV) = tr(VU) for arbitrary infinite matrices U and V. The determinant of an infinite matrix is defined through

$$\ln (det\mathit{U})=\text{tr}(\ln \mathit{U}).$$ 2.5

Again, this is a formal definition which could result in a divergence problem. But if we restrict ourselves to the infinite matrix 1 + U, where 1 is the identity infinite matrix and U is an infinite matrix involving the projection matrix O, the determinant is well-defined, since the right hand side of (2.5), namely

$$\text{tr}[\ln (1+\mathit{U})]=\text{tr}\left[\sum _{i=1}^{\mathrm{\infty }}\frac{{(-1)}^{i-1}}{i}{\mathit{U}}^i\right]=\sum _{i=1}^{\mathrm{\infty }}\frac{{(-1)}^{i-1}}{i}\text{tr}({\mathit{U}}^i),$$

has terms of convergent traces. We have the well-known Weinstein–Aronszajn formulas for the determinant. For instance, the following identities hold:

$$\begin{array}{r}det(1+\mathit{U}\mathit{O}\mathit{V})=1+{(\mathit{V}\mathit{U})}_{0,0},det(1+\mathit{U}(\mathit{O}\mathbf{\Lambda }-{\hspace{0.17em}}^t\mathbf{\Lambda }\mathit{O})\mathit{V})=det\left(\begin{array}{cc}1+{(\mathbf{\Lambda }\mathit{V}\mathit{U})}_{0,0}& -{(\mathbf{\Lambda }\mathit{V}{\hspace{0.17em}}^t\mathbf{\Lambda })}_{0,0}\\ {(\mathit{V}\mathit{U})}_{0,0}& 1-{(\mathit{V}\mathit{U}{\hspace{0.17em}}^t\mathbf{\Lambda })}_{0,0}\end{array}\right).\end{array}$$

which are the cases of rank 1 and rank 2.

We provide the formal structure of the DL framework. The starting point is a linear integral equation

$${\mathit{u}}_k+{\iint }_D\mathrm{d}\zeta (l,l^{\prime }){\rho }_k{\mathrm{\Omega }}_{k,l^{\prime }}{\sigma }_{l^{\prime }}{\mathit{u}}_l={\rho }_k{\mathit{c}}_k,$$ 2.6

in which the wave function u_k is an infinite-dimensional column vector having its i-th component $u_k^{(i)}$ (for $i\in \mathbb{Z}$) being a function of the lattice variables n_j and the lattice parameters p_j for j = 1, 2, … as well as the spectral variable k, Ω_k,l′ is the kernel of the linear integral equation, depending on the spectral variables k and l′, dζ and D are the measure and the domain for integration, and ρ_k and σ_l′ are the so-called plane wave factors depending on the discrete variables n_j and lattice parameters p_j as well as their respective spectral variables k and l′. Furthermore, to derive equations in the given choice of independent variables (which could be either the discrete variables n_j or the continuous variables p_j) we assume that the measure is independent of these chosen variables.

We present the infinite matrix representation of (2.6). We first introduce the infinite matrix Ω defined through

$${\mathrm{\Omega }}_{k,k^{\prime }}={\hspace{0.17em}}^t{\mathit{c}}_{k^{\prime }}\mathbf{\Omega }\hspace{0.17em}{\mathit{c}}_k.$$ 2.7

Following from the properties of the index-raising matrices and the projection matrix listed above, we can observe that Ω is actually an infinite matrix composed of Λ, ^tΛ and O, which relies on the precise expression of Ω_k,k′, namely it is an infinite matrix representation of the Cauchy kernel of the linear integral equation (2.6). If we replace Ω_k,l′ with the help of (2.7), the linear integral equation is reformulated as

$${\mathit{u}}_k=(1-\mathit{U}\mathbf{\Omega }){\mathit{c}}_k{\rho }_k,$$ 2.8

where the infinite matrix U is given by

$$\mathit{U}\doteq {\iint }_D\mathrm{d}\zeta (k,k^{\prime }){\mathit{u}}_k{\hspace{0.17em}}^t{\mathit{c}}_{k^{\prime }}{\sigma }_{k^{\prime }}.$$ 2.9

Next, we consider the infinite matrix representation of the plane wave factors and introduce an infinite matrix

$$\mathit{C}\doteq {\iint }_D\mathrm{d}\zeta (k,k^{\prime }){\rho }_k{\mathit{c}}_k{\hspace{0.17em}}^t{\mathit{c}}_{k^{\prime }}{\sigma }_{k^{\prime }}.$$ 2.10

The key characteristic of C is the product of the two plane wave factors, i.e. ρ_kσ_k′, which we normally refer to as the so-called effective plane wave factor. By acting the operation ${\iint }_D\mathrm{d}\zeta (k,k^{\prime })\cdot {\hspace{0.17em}}^t{\mathit{c}}_{k^{\prime }}{\sigma }_{k^{\prime }}$ on equation (2.10), we obtain

$$\mathit{U}=(1-\mathit{U}\mathbf{\Omega })\mathit{C},\text{or equivalently}\hspace{0.17em}\mathit{U}=\mathit{C}{(1+\mathbf{\Omega }\mathit{C})}^{-1}.$$ 2.11

The idea of the DL approach is to associate a nonlinear equation with a linear integral equation in the form of (2.6). Once the plane wave factors, the Cauchy kernel and the measure are given, the corresponding class of nonlinear integrable systems is fully determined. To be more precise, for a certain class of nonlinear integrable equations, the infinite matrix C describes the linear dispersion, the infinite matrix Ω together with the measure governs the nonlinear structure of the corresponding integrable models, and u_k and U are corresponding to the wave function in the Lax pair and the nonlinear potential, respectively.

3. Derivation of (1.4a), (1.4b) and (1.4c)

In the AKP class, the discrete plane wave factors ρ_k and σ_k′ are given by

$${\rho }_k=\prod _{i=1}^{\mathrm{\infty }}{(p_i+k)}^{n_i}\text{and}{\sigma }_{k^{\prime }}=\prod _{i=1}^{\mathrm{\infty }}{(p_i-k^{\prime })}^{-n_i},$$ 3.1

respectively. Substituting these into the infinite matrix C defined by (2.10) and considering the evolutions with regard to the discrete variables and lattice parameters, we obtain dynamical evolutions as follows:

$$\begin{array}{rl}\hspace{0.17em}& ({\mathrm{T}}_{n_j}\mathit{C})(p_j-{\hspace{0.17em}}^t\mathbf{\Lambda })=(p_j+\mathbf{\Lambda })\mathit{C},\end{array}$$ 3.2a

$$\begin{array}{rl}\hspace{0.17em}& {\mathrm{\partial }}_{p_j}\mathit{C}=n_j(\frac{1}{p_j+\mathbf{\Lambda }}\mathit{C}-\mathit{C}\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }})\end{array}$$ 3.2b

$$\begin{array}{rl}\text{and}& ({\mathrm{T}}_a^{-1}{\mathrm{T}}_b\mathit{C})\frac{b-{\hspace{0.17em}}^t\mathbf{\Lambda }}{a-{\hspace{0.17em}}^t\mathbf{\Lambda }}=\frac{b+\mathbf{\Lambda }}{a+\mathbf{\Lambda }}\mathit{C}.\end{array}$$ 3.2c

Below we provide the derivation of (3.2b). By differentiating ρ_kσ_k′ with respect to p_j, we have

$$\begin{array}{rl}{\mathrm{\partial }}_{p_j}{\rho }_k{\sigma }_{k^{\prime }}& ={\mathrm{\partial }}_{p_j}\prod _{i=1}^{\mathrm{\infty }}{\left(\frac{p_i+k}{p_i-k^{\prime }}\right)}^{n_i}\\ \hspace{0.17em}& =n_j(\frac{1}{p_j+k}-\frac{1}{p_j-k^{\prime }})\prod _{i=1}^{\mathrm{\infty }}{\left(\frac{p_i+k}{p_i-k^{\prime }}\right)}^{n_i}\\ \hspace{0.17em}& =n_j(\frac{1}{p_j+k}-\frac{1}{p_j-k^{\prime }}){\rho }_k{\sigma }_{k^{\prime }},\end{array}$$

and thus, the same operation on C gives rise to

$$\begin{array}{rl}{\mathrm{\partial }}_{p_j}\mathit{C}& ={\iint }_D\mathrm{d}\zeta (k,k^{\prime }){\mathit{c}}_k{\mathrm{\partial }}_{p_j}({\rho }_k{\sigma }_{k^{\prime }}){\hspace{0.17em}}^t{\mathit{c}}_{k^{\prime }}={\iint }_D\mathrm{d}\zeta (k,k^{\prime }){\mathit{c}}_k{\mathrm{\partial }}_{p_j}({\rho }_k{\sigma }_{k^{\prime }}){\hspace{0.17em}}^t{\mathit{c}}_{k^{\prime }}\\ \hspace{0.17em}& =n_j({\iint }_D\mathrm{d}\zeta (k,k^{\prime })\frac{1}{p_j+k}{\mathit{c}}_k{\rho }_k{\sigma }_{k^{\prime }}{\hspace{0.17em}}^t{\mathit{c}}_{k^{\prime }}-{\iint }_D\mathrm{d}\zeta (k,k^{\prime }){\mathit{c}}_k{\rho }_k{\sigma }_{k^{\prime }}{\hspace{0.17em}}^t{\mathit{c}}_{k^{\prime }}\frac{1}{p_j-k^{\prime }})\\ \hspace{0.17em}& =n_j(\frac{1}{p_j+\mathbf{\Lambda }}\mathit{C}-\mathit{C}\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}),\end{array}$$

where in the last step we have made use of the property of the operation of the index-raising matrices on c_k and ^tc_k′ given in §2. The other two equations are derived similarly.

The Cauchy kernel in the linear integral equation for the discrete AKP equation takes the form of

$${\mathrm{\Omega }}_{k,k^{\prime }}=\frac{1}{k+k^{\prime }},$$ 3.3

and in this case we have no further requirement for the measure dζ(k, k′) and the domain D, namely they are arbitrary. According to (2.7), we can observe that in this case $\mathbf{\Omega }=-\sum _{i=0}^{\mathrm{\infty }}{(-{\hspace{0.17em}}^t\mathbf{\Lambda })}^{-i-1}\mathit{O}{\mathbf{\Lambda }}^i$; in other words, it satisfies

$$\mathbf{\Omega }\mathbf{\Lambda }+{\hspace{0.17em}}^t\mathbf{\Lambda }\mathbf{\Omega }=\mathit{O}.$$ 3.4

Equation (3.4) can also be written in other forms. Here, we reformulate it in the following forms whose left-hand sides are compatible with (3.2):

$$\begin{array}{rl}\hspace{0.17em}& \mathbf{\Omega }(p_j+\mathbf{\Lambda })+(p_j-{\hspace{0.17em}}^t\mathbf{\Lambda })\mathbf{\Omega }=\mathit{O},\end{array}$$ 3.5a

$$\begin{array}{rl}\hspace{0.17em}& \mathbf{\Omega }\frac{1}{p_j+\mathbf{\Lambda }}-\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathbf{\Omega }=\frac{1}{-p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathit{O}\frac{1}{p_j+\mathbf{\Lambda }}\end{array}$$ 3.5b

$$\begin{array}{rl}\text{and}& \mathbf{\Omega }\frac{b+\mathbf{\Lambda }}{a+\mathbf{\Lambda }}-\frac{b-{\hspace{0.17em}}^t\mathbf{\Lambda }}{a-{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathbf{\Omega }=-(a-b)\frac{1}{-a+{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathit{O}\frac{1}{a+\mathbf{\Lambda }}.\end{array}$$ 3.5c

Equation (3.5) together with (3.2) will provide the dynamical evolutions of the infinite matrix U as follows:

$$\begin{array}{rl}\hspace{0.17em}& ({\mathrm{T}}_{n_j}\mathit{U})(p_j-{\hspace{0.17em}}^t\mathbf{\Lambda })=(p_j+\mathbf{\Lambda })\mathit{U}-({\mathrm{T}}_{n_j}\mathit{U})\mathit{O}\mathit{U},\end{array}$$ 3.6a

$$\begin{array}{rl}\hspace{0.17em}& {\mathrm{\partial }}_{p_j}\mathit{U}=n_j(\frac{1}{p_j+\mathbf{\Lambda }}\mathit{U}-\mathit{U}\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}-\mathit{U}\frac{1}{-p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathit{O}\frac{1}{p_j+\mathbf{\Lambda }}\mathit{U})\end{array}$$ 3.6b

$$\begin{array}{rl}\text{and}& ({\mathrm{T}}_a{\mathrm{T}}_b^{-1}\mathit{U})\frac{b-{\hspace{0.17em}}^t\mathbf{\Lambda }}{a-{\hspace{0.17em}}^t\mathbf{\Lambda }}=\frac{a+\mathbf{\Lambda }}{b+\mathbf{\Lambda }}\mathit{U}+(a-b)({\mathrm{T}}_a^{-1}{\mathrm{T}}_b\mathit{U})\frac{1}{-a+{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathit{O}\frac{1}{a+\mathbf{\Lambda }}\mathit{U}.\end{array}$$ 3.6c

Again we only give the derivation of (3.6b) below and skip that of the other two, as the procedure is similar. Note that the infinite matrix U obeys (2.11) in the formal structure of the DL. Calculating the derivative of U with respect to p_j, we obtain

$${\mathrm{\partial }}_{p_j}\mathit{U}=-({\mathrm{\partial }}_{p_j}\mathit{U})\mathbf{\Omega }\mathit{C}+(1-\mathit{U}\mathbf{\Omega })({\mathrm{\partial }}_{p_j}\mathit{C}),$$

which can equivalently be rewritten as

$$\begin{array}{rl}({\mathrm{\partial }}_{p_j}\mathit{U})(1+\mathbf{\Omega }\mathit{C})& =n_j(1-\mathit{U}\mathbf{\Omega })(\frac{1}{p_j+\mathbf{\Lambda }}\mathit{C}-\mathit{C}\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }})\\ \hspace{0.17em}& =n_j[\frac{1}{p_j+\mathbf{\Lambda }}\mathit{C}-\mathit{U}\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}-\mathit{U}\mathbf{\Omega }\frac{1}{p_j+\mathbf{\Lambda }}\mathit{C}]\\ \hspace{0.17em}& =n_j[\frac{1}{p_j+\mathbf{\Lambda }}\mathit{C}-\mathit{U}\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}(1+\mathbf{\Omega }\mathit{C})-\mathit{U}\frac{1}{-p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathit{O}\frac{1}{p_j+\mathbf{\Lambda }}\mathit{C}],\end{array}$$

where in the first and third equalities we have used (3.2b) and (3.5b), respectively. Multiplying the above equation by (1 + ΩC)⁻¹ immediately gives rise to (3.6b).

Next, we introduce the tau function in this class, which is defined as

$$\tau \doteq det(1+\mathbf{\Omega }\mathit{C}),$$ 3.7

where 1 denotes the infinite unit matrix. As we have mentioned in §2, the determinant of an infinite matrix should be understood as the formal expansion of exp{tr[ln(1 + ΩC)]}. Since $\mathbf{\Omega }=-\sum _{i=0}^{\mathrm{\infty }}{(-{\hspace{0.17em}}^t\mathbf{\Lambda })}^{-i-1}\mathit{O}{\mathbf{\Lambda }}^i$ involves O in every term, the trace action is always convergent. For convenience, we introduce quantities

$$\begin{array}{rl}\hspace{0.17em}& u\doteq {(U)}_{0,0},V_a\doteq 1-{\left(\mathit{U}\frac{1}{a+{\hspace{0.17em}}^t\mathbf{\Lambda }}\right)}_{0,0},\\ \hspace{0.17em}& W_a\doteq 1-{\left(\frac{1}{a+\mathbf{\Lambda }}\mathit{U}\right)}_{0,0}\text{and}{S}_{a,b}\doteq {\left(\frac{1}{a+\mathbf{\Lambda }}\mathit{U}\frac{1}{b+{\hspace{0.17em}}^t\mathbf{\Lambda }}\right)}_{0,0}.\end{array}$$

The tau function satisfies the dynamical evolutions with respect to the discrete variables and the lattice parameters as follows:

$$\begin{array}{rl}\hspace{0.17em}& \frac{{\mathrm{T}}_{n_j}\tau }{\tau }=1-{\left(\mathit{U}\frac{1}{-p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}\right)}_{0,0}=V_{-p_j},\frac{{\mathrm{T}}_{n_j}^{-1}\tau }{\tau }=1-{\left(\frac{1}{p_j+\mathbf{\Lambda }}\mathit{U}\right)}_{0,0}=W_{p_j},\end{array}$$ 3.8a

$$\begin{array}{rl}\hspace{0.17em}& {\mathrm{\partial }}_{p_j}\ln \tau =n_j{S}_{p_j,-p_j},\end{array}$$ 3.8b

$$\begin{array}{rl}\hspace{0.17em}& \frac{{\mathrm{T}}_a^{-1}{\mathrm{T}}_b\tau }{\tau }=1-(a-b){\left(\frac{1}{a+\mathbf{\Lambda }}\mathit{U}\frac{1}{-b+{\hspace{0.17em}}^t\mathbf{\Lambda }}\right)}_{0,0}=1-(a-b)S_{a,-b}\end{array}$$ 3.8c

$$\begin{array}{rl}\text{and}& (p_i-p_j)\frac{\tau ({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}\tau )}{({\mathrm{T}}_{n_i}\tau )({\mathrm{T}}_{n_j}\tau )}=p_i-p_j+{\mathrm{T}}_{n_j}u-{\mathrm{T}}_{n_i}u.\end{array}$$ 3.8d

These equations are proven through direct computation in terms of infinite matrices. For instance, calculating the derivative of lnτ with respect to p_j yields

$$\begin{array}{rl}{\mathrm{\partial }}_{p_j}\ln \tau & ={\mathrm{\partial }}_{p_j}\ln [det(1+\mathbf{\Omega }\mathit{C})]={\mathrm{\partial }}_{p_j}\text{tr}[\ln (1+\mathbf{\Omega }\mathit{C})]\\ \hspace{0.17em}& =\text{tr}[{\mathrm{\partial }}_{p_j}\ln (1+\mathbf{\Omega }\mathit{C})]=\text{tr}[{(1+\mathbf{\Omega }\mathit{C})}^{-1}\mathbf{\Omega }({\mathrm{\partial }}_{p_j}\mathit{C})]\\ \hspace{0.17em}& =n_j\text{tr}\left[{(1+\mathbf{\Omega }\mathit{C})}^{-1}\mathbf{\Omega }(\frac{1}{p_j+\mathbf{\Lambda }}\mathit{C}-\mathit{C}\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }})\right]\\ \hspace{0.17em}& =n_j\text{tr}\left[\mathit{C}{(1+\mathbf{\Omega }\mathit{C})}^{-1}(\mathbf{\Omega }\frac{1}{p_j+\mathbf{\Lambda }}-\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathbf{\Omega })\right],\end{array}$$

where in the last step the property of the cyclic permutation of the trace operation is used. With the help of (2.11) and (3.5b), this equation is reformulated as

$$\begin{array}{rl}{\mathrm{\partial }}_{p_j}\ln \tau & =n_j\text{tr}\left(\mathit{U}\frac{1}{-p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathit{O}\frac{1}{p_j+\mathbf{\Lambda }}\right)\\ \hspace{0.17em}& =n_j\text{tr}\left(\mathit{O}\frac{1}{p_j+\mathbf{\Lambda }}\mathit{U}\frac{1}{-p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}\right)=n_j{\left(\frac{1}{p_j+\mathbf{\Lambda }}\mathit{U}\frac{1}{-p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}\right)}_{0,0},\end{array}$$

namely ${\mathrm{\partial }}_{p_j}\ln \tau =n_j{S}_{p_j,-p_j}$. To prove equation (3.8a) and (3.8c), one needs to use (3.2a) and (3.5a) and (3.2c) and (3.5c), respectively. Since the idea of the proofs is similar to that of (3.8a), we skip them here. While equation (3.8d) is proven based on (3.8a) and (3.6), and it is a widely known relation which describes bilinear transformation for the discrete KP equation (e.g. [8]).

Equations listed in (3.6) are the key formulas in the DL scheme to construct closed-form integrable equations, and they together with (3.8) can produce the bilinear equations in the AKP class. In [8], the well-known HM equation, i.e. equation (1.1), is derived from (3.6a) and (3.8a). Here, we start from the dynamical relations in terms of p_j to construct the (2+2)-dimensional non-autonomous DΔE in this class. Considering ${(a+\mathbf{\Lambda })}^{-1}\text{(3.6b)}{(b+{\hspace{0.17em}}^t\mathbf{\Lambda })}^{-1}$(3.6b) and taking the (0, 0)-entry, we have the following equation for the quantity S_a,b:

$${\mathrm{\partial }}_{p_j}{S}_{a,b}=n_j[(\frac{1}{p_j-a}-\frac{1}{p_j+b})S_{a,b}-\frac{1}{p_j-a}{S}_{p_j,b}+\frac{1}{p_j+b}{S}_{a,-p_j}-S_{a,-p_j}{S}_{p_j,b}].$$

By setting a = p_i and b = −p_i, we reach to

$${\mathrm{\partial }}_{p_j}{S}_{p_i,-p_i}=\frac{n_j}{{(p_i-p_j)}^2}[(1-(p_j-p_i)S_{p_j,-p_i})(1-(p_i-p_j)S_{p_i,-p_j})-1],$$

which only involves the S-variable. If we replace all the S-variables in this equation by the tau function via (3.8) and note the identity

$${\mathrm{\partial }}_{p_i}{\mathrm{\partial }}_{p_j}\ln \tau =\frac{{\mathrm{D}}_{p_i}{\mathrm{D}}_{p_j}\tau \cdot \tau }{2{\tau }^2}$$

for Hirota’s bilinear operator D_·, the bilinear equation (1.4a) arises. This equation takes the form of a non-autonomous version of the two-dimensional (2D) Toda equation, if we think of the discrete shift operations ${\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}^{-1}$ and ${\mathrm{T}}_{n_i}^{-1}{\mathrm{T}}_{n_j}$ as the forward and backward shifts along the skew direction on the lattice. The difference is that here the lattice parameters p_i and p_j act as the independent variables, while in the two-dimensional Toda lattice, the bilinear derivatives are with respect to the continuous flow variables x₁ and x₋₁ (cf. [33]), which leads to an autonomous equation.

From the scheme, we can also construct two more non-autonomous semi-discrete equations of τ. Taking ${[\text{(3.6b)}{(a+{\hspace{0.17em}}^t\mathbf{\Lambda })}^{-1}]}_{0,0}$(3.6b), one obtains

$${\mathrm{\partial }}_{p_j}{V}_a=-n_j[V_{-p_j}{S}_{p_j,a}+\frac{1}{a+p_j}(V_a-V_{-p_j})],$$

and its a = −p_i case gives rise to

$${\mathrm{\partial }}_{p_j}{V}_{-p_i}=\frac{n_j}{p_i-p_j}[V_{-p_i}-V_{-p_j}(1-(p_j-p_i)S_{p_j,-p_i})].$$

Substituting the V- and S-variables with the tau function with the help of (3.8), we are able to construct the bilinear equation (1.4b).

Moreover, if we consider the equation for ${\mathrm{\partial }}_{p_j}u$ by evaluating the (0, 0)-entry of (3.6b), the following equation shows up:

$${\mathrm{\partial }}_{p_j}u=n_j(1-V_{-p_j}{W}_{p_j})=n_j[1-\frac{({\mathrm{T}}_{n_j}\tau )({\mathrm{T}}_{n_j}^{-1}\tau )}{{\tau }^2}],$$

where formula (3.8) is used for the second equality, and therefore, we obtain from (3.8d) that

$${\mathrm{\partial }}_{p_j}[(p_i-p_j)\frac{\tau ({\mathrm{T}}_{n_i}{\mathrm{T}}_{n_j}\tau )}{({\mathrm{T}}_{n_i}\tau )({\mathrm{T}}_{n_j}\tau )}]={\mathrm{\partial }}_{p_j}(p_i-p_j+{\mathrm{T}}_{n_j}u-{\mathrm{T}}_{n_i}u)=-1+{\mathrm{T}}_{n_j}({\mathrm{\partial }}_{p_j}u)-{\mathrm{T}}_{n_i}({\mathrm{\partial }}_{p_j}u).$$

Finally, by substituting u with τ, a closed-form quartic equation of τ is constructed, which takes the form of equation (1.4c).

4. Derivation of (1.5a) and (1.5b)

We select the following plane wave factors for the BKP class:

$${\rho }_k=\prod _{i=1}^{\mathrm{\infty }}{\left(\frac{p_i+k}{p_i-k}\right)}^{n_i},{\sigma }_{k^{\prime }}={\rho }_{k^{\prime }}=\prod _{i=1}^{\mathrm{\infty }}{\left(\frac{p_i+k^{\prime }}{p_i-k^{\prime }}\right)}^{n_i},$$ 4.1

which should be understood as discrete odd flows, compared with (3.1) in the AKP case. In this case, it is observed that the infinite matrix C defined in (2.10) evolves with regard to the discrete variables and the lattice parameters in the following way:

$$({\mathrm{T}}_{n_j}\mathit{C})\frac{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}=\frac{p_j+\mathbf{\Lambda }}{p_j-\mathbf{\Lambda }}\mathit{C}$$ 4.2a

and

$${\mathrm{\partial }}_{p_j}\mathit{C}=n_j[(\frac{1}{p_j+\mathbf{\Lambda }}-\frac{1}{p_j-\mathbf{\Lambda }})\mathit{C}+\mathit{C}(\frac{1}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}-\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }})].$$ 4.2b

The derivation of these formulas is straightforward, as they follow from the formal definition of C, namely (2.10).

The Cauchy kernel in the BKP case takes the form of

$${\mathrm{\Omega }}_{k,k^{\prime }}=\frac{1}{2}\frac{k-k^{\prime }}{k+k^{\prime }},$$ 4.3

which implies that the corresponding Ω satisfies

$$\mathbf{\Omega }\mathbf{\Lambda }+{\hspace{0.17em}}^t\mathbf{\Lambda }\mathbf{\Omega }=\frac{1}{2}(\mathbf{\Omega }\mathbf{\Lambda }-{\hspace{0.17em}}^t\mathbf{\Lambda }\mathit{O}).$$ 4.4

For the purpose of looking for compatible relations of Ω with the equations listed in (4.2), we reformulate (4.4) as

$$\mathbf{\Omega }\frac{p_j+\mathbf{\Lambda }}{p_j-\mathbf{\Lambda }}-\frac{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathbf{\Omega }=p_j\frac{1}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}(\mathit{O}\mathbf{\Lambda }-{\hspace{0.17em}}^t\mathbf{\Lambda }\mathit{O})\frac{1}{p_j-\mathbf{\Lambda }}$$ 4.5a

and

$$\begin{array}{rl}\hspace{0.17em}& \mathbf{\Omega }(\frac{1}{p_j+\mathbf{\Lambda }}-\frac{1}{p_j-\mathbf{\Lambda }})+(\frac{1}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}-\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }})\mathbf{\Omega }\\ \hspace{0.17em}& =-\frac{1}{2}\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}(\mathit{O}\mathbf{\Lambda }-{\hspace{0.17em}}^t\mathbf{\Lambda }\mathit{O})\frac{1}{p_j+\mathbf{\Lambda }}-\frac{1}{2}\frac{1}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}(\mathit{O}\mathbf{\Lambda }-{\hspace{0.17em}}^t\mathbf{\Lambda }\mathit{O})\frac{1}{p_j-\mathbf{\Lambda }},\end{array}$$ 4.5b

respectively.

Equations (4.2) and (4.5) are the fundamental relations to construct the dynamical evolutions of the infinite matrix U. By considering ${\mathrm{T}}_{n_j}U$ and ${\mathrm{\partial }}_{p_j}\mathit{U}$ in (2.11), some straightforward calculations yield the following equations:

$$({\mathrm{T}}_{n_j}\mathit{U})\frac{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}=\frac{p_j+\mathbf{\Lambda }}{p_j-\mathbf{\Lambda }}\mathit{U}-p_j({\mathrm{T}}_{n_j}\mathit{U})\frac{1}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}(\mathit{O}\mathbf{\Lambda }-{\hspace{0.17em}}^t\mathbf{\Lambda }\mathit{O})\frac{1}{p_j-\mathbf{\Lambda }}\mathit{U}$$ 4.6a

and

$$\begin{array}{rl}{\mathrm{\partial }}_{p_j}\mathit{U}& =n_j[(\frac{1}{p_j+\mathbf{\Lambda }}-\frac{1}{p_j-\mathbf{\Lambda }})\mathit{U}+\mathit{U}(\frac{1}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}-\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }})\\ \hspace{0.17em}& +\frac{1}{2}\mathit{U}\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}(\mathit{O}\mathbf{\Lambda }-{\hspace{0.17em}}^t\mathbf{\Lambda }\mathit{O})\frac{1}{p_j+\mathbf{\Lambda }}\mathit{U}+\frac{1}{2}\mathit{U}\frac{1}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}(\mathit{O}\mathbf{\Lambda }-{\hspace{0.17em}}^t\mathbf{\Lambda }\mathit{O})\frac{1}{p_j-\mathbf{\Lambda }}\mathit{U}],\end{array}$$ 4.6b

with the help of (4.2) and (4.5).

Besides the different kernel and plane wave factors, we also have to impose certain restrictions on the integration measure and the integration domain for the BKP equation. We require that the integration domain D is symmetric in terms of the spectral variables k and k′, and then the measure dζ(k, k′) is antisymmetric, i.e.

$$\mathrm{d}\zeta (k^{\prime },k)=-\mathrm{d}\zeta (k,k^{\prime }).$$ 4.7

These will together result in

$${\hspace{0.17em}}^t\mathit{C}={\iint }_D\mathrm{d}\zeta (k,k^{\prime }){\rho }_k{\mathit{c}}_{k^{\prime }}{\hspace{0.17em}}^t{\mathit{c}}_k{\rho }_{k^{\prime }}=-{\iint }_D\mathrm{d}\zeta (k^{\prime },k){\rho }_{k^{\prime }}{\mathit{c}}_{k^{\prime }}{\hspace{0.17em}}^t{\mathit{c}}_k{\rho }_k=-\mathit{C};$$

in other words, we have an antisymmetric infinite matrix C in the discrete BKP. Note that the kernel given in (4.3) is also antisymmetric, or equivalently ^tΩ = −Ω. We can deduce from (2.11) that in this case the infinite matrix U obeys the antisymmetry property

$${\hspace{0.17em}}^t\mathit{U}=-\mathit{U}.$$ 4.8

Next, we present the dynamical evolutions of the tau function. For convenience, we introduce quantities

$$V_a\doteq 1-{\left(\mathit{U}\frac{a}{a-{\hspace{0.17em}}^t\mathbf{\Lambda }}\right)}_{0,0},W_a\doteq 1+{\left(\frac{a}{a-\mathbf{\Lambda }}\mathit{U}\right)}_{0,0}\text{and}{S}_{a,b}\doteq {\left(\frac{a}{a-\mathbf{\Lambda }}\mathit{U}\frac{b}{b-{\hspace{0.17em}}^t\mathbf{\Lambda }}\right)}_{0,0}.$$

Due to the antisymmetry property of the infinite matrix U, it is obvious to see that

$$V_a=W_a\text{and}{S}_{a,b}=-S_{b,a}.$$

We define the tau function in this class by

$${\tau }^2=det(1+\mathbf{\Omega }\mathit{C}),$$ 4.9

since the antisymmetry of O and C will eventually make the above determinant a perfect square, namely the tau function itself is corresponding to a Pfaffian. We now consider the evolution of the tau function with respect to n_j, and this gives us

$$\begin{array}{rl}{\mathrm{T}}_{n_j}{\tau }^2& =det[1+\mathbf{\Omega }({\mathrm{T}}_{n_j}\mathit{C})]=det[1+\mathbf{\Omega }\mathit{C}+p_j\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}(\mathit{O}\mathbf{\Lambda }-{\hspace{0.17em}}^t\mathbf{\Lambda }\mathit{O})\frac{1}{p_j-\mathbf{\Lambda }}\mathit{C}]\\ \hspace{0.17em}& ={\tau }^{2}det\left[\begin{array}{cc}1+p{(\frac{\mathbf{\Lambda }}{p-\mathbf{\Lambda }}\mathit{U}\frac{1}{p-{\hspace{0.17em}}^t\mathbf{\Lambda }})}_{0,0}& -p{(\frac{\mathbf{\Lambda }}{p-\mathbf{\Lambda }}\mathit{U}\frac{{\hspace{0.17em}}^t\mathbf{\Lambda }}{p-{\hspace{0.17em}}^t\mathbf{\Lambda }})}_{0,0}\\ p{(\frac{1}{p-\mathbf{\Lambda }}\mathit{U}\frac{1}{p-{\hspace{0.17em}}^t\mathbf{\Lambda }})}_{0,0}& 1-p{\left(\frac{1}{p-\mathbf{\Lambda }}\mathit{U}\frac{{\hspace{0.17em}}^t\mathbf{\Lambda }}{p-{\hspace{0.17em}}^t\mathbf{\Lambda }}\right)}_{0,0}\end{array}\right]={\tau }^2{V}_{p_j}^2,\end{array}$$

where the rank 2 Weinstein–Aronszajn formula and equation (4.8) are used in the third and fourth equalities, respectively. Similarly, acting the backward shift on τ gives us ${\mathrm{T}}_{n_j}^{-1}{\tau }^2/\tau =V_{-p_j}^2$. In order to simplify this formula, we evaluate ${[\text{(4.6a)}]}_{0,0}$(4.6a), which gives us $V_{p_j}({\mathrm{T}}_{n_j}{V}_{-p_j})=1$. And thus, without loss of generality, we have

$$\frac{{\mathrm{T}}_{n_j}\tau }{\tau }=V_{p_j}\text{and}\frac{{\mathrm{T}}_{n_j}^{-1}\tau }{\tau }=V_{-p_j}.$$ 4.10a

Next, we calculate ${[p_i/(p_i-\mathbf{\Lambda })\text{(4.6a)}]}_{0,0}$(4.6a). The following equation is obtained:

$$1+2V_{p_j}({\mathrm{T}}_{n_j}{S}_{-p_i,-p_j})=\frac{p_i-p_j}{p_i+p_j}(V_{p_j}-V_{-p_i})+V_{p_j}({\mathrm{T}}_{n_j}{V}_{-p_i}).$$

This equation provides a way to express the S-variable by the V-variables, and consequently, the S-variable can be expressed by the tau function by making use of (4.10), and the formula is

$$S_{-p_i,-p_j}=\frac{1}{2}[\frac{{\mathrm{T}}_{n_i}^{-1}\tau -{\mathrm{T}}_{n_j}^{-1}\tau }{\tau }+\frac{p_i-p_j}{p_i+p_j}(1-\frac{{\mathrm{T}}_{n_i}^{-1}{\mathrm{T}}_{n_j}^{-1}\tau }{\tau })].$$ 4.10b

The expressions $S_{p_i,-p_j}$, $S_{-p_i,p_j}$ and $S_{p_i,p_j}$ in terms of the tau function can also be derived from the above equation, with the help of $(p_i,{\mathrm{T}}_{n_i})\leftrightarrow (-p_i,{\mathrm{T}}_{n_i}^{-1})$. Therefore, we have obtained the dynamical relations of τ in terms of S_a,b for arbitrary a + b ≠ 0. Furthermore, following the same idea of deriving (3.8b), we also have

$$2p_j{\mathrm{\partial }}_{p_j}\ln \tau =n_j(V_{p_j}-V_{-p_j}-2S_{p_j,-p_j})$$ 4.10c

for BKP. The equations listed in (4.10) establish the relations between the dynamics of the tau function and the V- and S-variables.

To construct equation (1.5a) in the DL framework, we compute ${[a/(a-\mathbf{\Lambda })\text{(4.6b)}]}_{0,0}$(4.6b) and ${[a/(a-\mathbf{\Lambda })\text{(4.6b)}b/(b-{\hspace{0.17em}}^t\mathbf{\Lambda })]}_{0,0}$(4.6b), which gives rise to

$$\begin{array}{rl}{p}_j{\mathrm{\partial }}_{p_j}{V}_a& =n_j[V_a(\frac{2a{p}_j}{a^2-p_j^2}-\frac{1}{2}{V}_{p_j}+\frac{1}{2}{V}_{-p_j})+V_{p_j}(\frac{1}{2}\frac{p_j+a}{p_j-a}+S_{a,-p_j})-V_{-p_j}(\frac{1}{2}\frac{p_j-a}{p_j+a}+S_{a,p_j})]\end{array}$$ 4.11a

and

$$\begin{array}{rl}{p}_j{\mathrm{\partial }}_{p_j}{S}_{a,b}& =n_j[\frac{1}{2}{S}_{p_j,b}(\frac{p_j+a}{p_j-a}-W_a)+\frac{1}{2}{S}_{-p_j,b}(\frac{a-p_j}{a+p_j}+W_a)\\ \hspace{0.17em}& +\frac{1}{2}{S}_{a,p_j}(\frac{p_j+b}{p_j-b}-V_b)+\frac{1}{2}{S}_{a,-p_j}(\frac{b-p_j}{b+p_j}+V_b)\\ \hspace{0.17em}& +S_{a,b}(\frac{2a{p}_j}{a^2-p_j^2}+\frac{2b{p}_j}{b^2-p_j^2})-S_{a,p_j}{S}_{-p_j,b}+S_{a,-p_j}{S}_{p_j,b}],\end{array}$$ 4.11b

respectively. These two equations allow us to derive the closed-form equations in terms of the tau function. Note that the logarithm derivative of τ with respect to p_j satisfies (4.10c), and hence we have

$$\begin{array}{rl}{p}_i{p}_j{\mathrm{\partial }}_{p_i}{\mathrm{\partial }}_{p_i}\ln \tau & =p_j{\mathrm{\partial }}_{p_j}(p_i{\mathrm{\partial }}_{p_i}\ln \tau )=p_j{\mathrm{\partial }}_{p_j}[n_i(V_{p_i}-V_{-p_i}-2S_{p_i,p_i})]\\ \hspace{0.17em}& =n_i[p_j({\mathrm{\partial }}_{p_j}{V}_{p_i})-p_j({\mathrm{\partial }}_{p_j}{V}_{-p_i})-2p_j({\mathrm{\partial }}_{p_j}{S}_{p_i,p_i})],\end{array}$$

in which the right-hand side only involves the derivatives of the V- and S-variables with respect to p_j, and they can be further replaced by terms of V- and S-variables without derivatives with the help of (4.11a) and (4.11a) for special a and b. Finally, equations (4.10a) and (4.10b) help us to express every term in the right-hand side of the above equation by the tau function. As a result, we derive (1.5a).

We can also derive an analogue of (1.4b) in the BKP class, from the DL scheme in this section. Setting a = p_i in (4.11a) yields

$$\begin{array}{rl}{p}_j{\mathrm{\partial }}_{p_j}{V}_{p_i}& =n_j[V_{p_i}(\frac{2p_i{p}_j}{p_i^2-p_j^2}-\frac{1}{2}{V}_{p_j}+\frac{1}{2}{V}_{-p_j})\\ \hspace{0.17em}& +V_{p_j}(\frac{1}{2}\frac{p_j+p_i}{p_j-p_i}+S_{p_i,-p_j})-V_{-p_j}(\frac{1}{2}\frac{p_j-p_i}{p_j+p_i}+S_{p_i,p_j})].\end{array}$$

Substituting V and S according to (4.10), we obtain (1.5b).

An interesting observation is that the right sides of (1.5a) and (1.5b) take the forms of Hirota’s discrete-time Toda equation (a five-point equation) [34] and the bilinear discrete KdV equation (a six-point equation), respectively (see also ch. 8 of [1]), although the parametrization here is entirely different.

5. Derivation of (1.6)

We choose the same plane wave factors as those of BKP in the CKP class, namely

$${\rho }_k=\prod _{i=1}^{\mathrm{\infty }}{\left(\frac{p_i+k}{p_i-k}\right)}^{n_i}\text{and}{\sigma }_{k^{\prime }}={\rho }_{k^{\prime }}=\prod _{i=1}^{\mathrm{\infty }}{\left(\frac{p_i+k^{\prime }}{p_i-k^{\prime }}\right)}^{n_i},$$ 5.1

and these provide the same dynamical relations for the infinite matrix C as follows:

$$({\mathrm{T}}_{n_j}\mathit{C})\frac{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}=\frac{p_j+\mathbf{\Lambda }}{p_j-\mathbf{\Lambda }}\mathit{C}$$ 5.2a

and

$${\mathrm{\partial }}_{p_j}\mathit{C}=n_j[(\frac{1}{p_j+\mathbf{\Lambda }}-\frac{1}{p_j-\mathbf{\Lambda }})\mathit{C}+\mathit{C}(\frac{1}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}-\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }})],$$ 5.2b

which describe the linear dispersion of the discrete CKP equation.

We select the Cauchy kernel

$${\mathrm{\Omega }}_{k,k^{\prime }}=\frac{1}{k+k^{\prime }}$$ 5.3

for the CKP class, which is the same as that in the AKP case, and therefore we still have the fundamental relation for Ω given by

$$\mathbf{\Omega }\mathbf{\Lambda }+{\hspace{0.17em}}^t\mathbf{\Lambda }\mathbf{\Omega }=\mathit{O}.$$ 5.4

Since the evolution of the infinite matrix C is governed by (5.2), we reformulate (5.4) and present the following equations:

$$\mathbf{\Omega }\frac{p_j+\mathbf{\Lambda }}{p_j-\mathbf{\Lambda }}-\frac{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathbf{\Omega }=2p_j\frac{1}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathit{O}\frac{1}{p_j-\mathbf{\Lambda }}$$ 5.5a

and

$$\mathbf{\Omega }(\frac{1}{p_j+\mathbf{\Lambda }}-\frac{1}{p_j-\mathbf{\Lambda }})+(\frac{1}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}-\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }})\mathbf{\Omega }=-\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathit{O}\frac{1}{p_j+\mathbf{\Lambda }}-\frac{1}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathit{O}\frac{1}{p_j-\mathbf{\Lambda }},$$ 5.5b

in order to derive the dynamical relations of U below.

Now, we construct the dynamical evolutions of the infinite matrix U. Acting the shift operator ${\mathrm{T}}_{n_j}$ on equation (2.11) and taking (5.2a), we have

$${\mathrm{T}}_{n_j}\mathit{U}=[1-({\mathrm{T}}_{n_j}\mathit{U})\mathbf{\Omega }]({\mathrm{T}}_{n_j}\mathit{C})=[1-({\mathrm{T}}_{n_j}\mathit{U})\mathbf{\Omega }]\frac{p_j+\mathbf{\Lambda }}{p_j-\mathbf{\Lambda }}\hspace{0.17em}\mathit{C}\hspace{0.17em}\frac{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}.$$

Note that the infinite matrix Ω obeys (5.5a). We can rewrite the above equation as

$$\begin{array}{rl}({\mathrm{T}}_{n_j}\mathit{U})\frac{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}& =\frac{p_j+\mathbf{\Lambda }}{p_j-\mathbf{\Lambda }}\mathit{C}-({\mathrm{T}}_{n_j}\mathit{U})\mathbf{\Omega }\frac{p_j+\mathbf{\Lambda }}{p_j-\mathbf{\Lambda }}\mathit{C}\\ \hspace{0.17em}& =\frac{p_j+\mathbf{\Lambda }}{p_j-\mathbf{\Lambda }}\mathit{C}-({\mathrm{T}}_{n_j}\mathit{U})(2p_j\frac{1}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathit{O}\frac{1}{p_j-\mathbf{\Lambda }}+\frac{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathbf{\Omega })\mathit{C},\end{array}$$

and this can be further simplified as

$$({\mathrm{T}}_{n_j}\mathit{U})\frac{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}=\frac{p_j+\mathbf{\Lambda }}{p_j-\mathbf{\Lambda }}\mathit{U}-2p_j({\mathrm{T}}_{n_j}\mathit{U})\frac{1}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathit{O}\frac{1}{p_j-\mathbf{\Lambda }}\mathit{U}.$$ 5.6a

Equation (5.6a) is the first dynamical relation we need and it describes how U evolves along the lattice directions n_j. The other dynamical relation we need is the one evolving with regard to p_j, and the derivation is very similar to those of (3.6b) and (4.6b), which reads

$$\begin{array}{rl}{\mathrm{\partial }}_{p_j}\mathit{U}\ast & =n_j[(\frac{1}{p_j+\mathbf{\Lambda }}-\frac{1}{p_j-\mathbf{\Lambda }})\mathit{U}+\mathit{U}(\frac{1}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}-\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }})\\ \hspace{0.17em}& +\mathit{U}\frac{1}{p_j-{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathit{O}\frac{1}{p_j+\mathbf{\Lambda }}\mathit{U}+\mathit{U}\frac{1}{p_j+{\hspace{0.17em}}^t\mathbf{\Lambda }}\mathit{O}\frac{1}{p_j-\mathbf{\Lambda }}\mathit{U}].\end{array}$$ 5.6b

As is similar to the BKP case, we also need to impose a certain constraint on the spectral variables k and k′. This is realized by setting the integration domain D symmetric and simultaneously requiring integration measure satisfying

$$\mathrm{d}\zeta (k^{\prime },k)=\mathrm{d}\zeta (k,k^{\prime }).$$ 5.7

Such a reduction results in a symmetric infinite matrix C because

$${\hspace{0.17em}}^t\mathit{C}={\iint }_D\mathrm{d}\zeta (k,k^{\prime }){\rho }_k{\mathit{c}}_{k^{\prime }}{\hspace{0.17em}}^t{\mathit{c}}_k{\rho }_{k^{\prime }}={\iint }_D\mathrm{d}\zeta (k^{\prime },k){\rho }_{k^{\prime }}{\mathit{c}}_{k^{\prime }}{\hspace{0.17em}}^t{\mathit{c}}_k{\rho }_k=\mathit{C}.$$

At the same time, we observe that the kernel (5.3) is symmetric in terms of k and k′, which implies that ^tΩ = Ω. The symmetry properties of C and Ω together guarantee that U from (2.11) satisfies

$${\hspace{0.17em}}^t\mathit{U}=\mathit{U},$$ 5.8

i.e. it is a symmetric infinite matrix.

During the derivation of (1.6) below, for convenience we introduce the variables

$$V_a\doteq 1-{\left(\mathit{U}\frac{1}{a+{\hspace{0.17em}}^t\mathbf{\Lambda }}\right)}_{0,0},W_a\doteq 1-{\left(\frac{1}{a+\mathbf{\Lambda }}\mathit{U}\right)}_{0,0}\text{and}{S}_{a,b}\doteq {\left(\frac{1}{a+\mathbf{\Lambda }}\mathit{U}\frac{1}{b+{\hspace{0.17em}}^t\mathbf{\Lambda }}\right)}_{0,0},$$ 5.9

which satisfy

$$V_a=W_a\text{and}{S}_{a,b}=S_{b,a},$$ 5.10

respectively, due to the symmetry condition (5.8). We define the tau function in the CKP class as

$$\tau =det(1+\mathbf{\Omega }\mathit{C}),$$ 5.11

and after a certain straightforward but relatively complex calculation based on (5.6a), we can derive

$$\frac{{\mathrm{T}}_{n_j}\tau }{\tau }=1+2p_j{S}_{-p_j,-p_j}\text{and}\frac{{\mathrm{T}}_{n_j}^{-1}\tau }{\tau }=1-2p_j{S}_{p_j,p_j},$$ 5.12a

as well as

$${[1-(p_i+p_j)S_{p_i,p_j}]}^2=\frac{{(p_i+p_j)}^2({\mathrm{T}}_{n_i}^{-1}\tau )({\mathrm{T}}_{n_j}^{-1}\tau )-{(p_i-p_j)}^2\tau ({\mathrm{T}}_{n_i}^{-1}{\mathrm{T}}_{n_j}^{-1}\tau )}{4p_i{p}_j{\tau }^2}.$$ 5.12b

Similar to the BKP case, the $S_{-p_i,p_j}$, $S_{p_i,-p_j}$ and $S_{-p_i,-p_j}$ analogues of (5.12b) follow from the interchange relation $(p_i,{\mathrm{T}}_{n_i})\leftrightarrow (-p_i,{\mathrm{T}}_{n_j}^{-1}$. Moreover, the dynamics of the tau function with respect to the lattice parameter p_j is derived by carrying the same calculation in the derivation of (3.8b), which establishes the link between τ and S and takes the form of

$${\mathrm{\partial }}_{p_j}\ln \tau =2n_j{S}_{p_j,-p_j}.$$ 5.12c

Note from (5.12) that the tau function is connected with the S-variable. We therefore multiply (5.6b) by (a + Λ)⁻¹ from the left and (b + ^tΛ)⁻¹ from the right simultaneously and take the (0, 0)-entry, and as a result, an equation for S_a,b arises in the form of

$$\begin{array}{rl}{\mathrm{\partial }}_{p_j}{S}_{a,b}& =n_j[\frac{1}{p_j-a}(S_{a,b}-S_{p_j,b})+\frac{1}{p_j+a}(S_{-p_j,b}-S_{a,b})\\ \hspace{0.17em}& +\frac{1}{b-p_j}(S_{a,p_j}-S_{a,b})+\frac{1}{p_j+b}(S_{a,-p_j}-S_{a,b})-S_{a,-p_j}{S}_{p_j,b}-S_{a,p_j}{S}_{-p_j,b}].\end{array}$$

By setting a = p_i and b = −p_i, it is rewritten as

$$\begin{array}{rl}{\mathrm{\partial }}_{p_j}{S}_{p_i,-p_i}& =\frac{n_j}{{(p_i+p_j)}^2}[(1-(p_i+p_j)S_{p_i,p_j})(1+(p_i+p_j)S_{-p_i,-p_j})-1]\\ \hspace{0.17em}& +\frac{n_j}{{(p_i-p_j)}^2}[(1-(p_i-p_j)S_{p_i,-p_j})(1-(p_j-p_i)S_{-p_i,p_j})-1].\end{array}$$

It is then possible to replace the S-variable by τ through (5.12a) and (5.12b) in the above equation. As a result, it leads to the non-autonomous differential-difference CKP equation (1.6). We note that this equation is still in the form of the (2 + 2)-dimensional Toda-type, but here the bilinearity is broken (though the equation is still quadratic) compared with (1.4a) and (1.5a) since the square root operation is involved. This is not surprising as we have seen the discrete CKP equation (1.3) is a quartic equation. Equation (1.6) can alternatively be written as an eighth-power equation

$$\begin{array}{rl}\hspace{0.17em}& {\{{[\frac{p_i{p}_j{(p_i^2-p_j^2)}^2}{n_i{n}_j}{\mathrm{D}}_{p_i}{\mathrm{D}}_{p_j}\tau \cdot \tau +8p_i{p}_j(p_i^2+p_j^2){\tau }^2]}^2-{(p_i-p_j)}^{4}EF-{(p_i+p_j)}^{4}GH\}}^2\\ \hspace{0.17em}& =4{(p_i^2-p_j^2)}^{4}EFGH\end{array}$$

by taking the square twice in order to eliminate the square root in the expression, where E, F, G and H are already given in §1 (1.3).

6. Remarks on soliton solutions, multi-dimensional consistency

and higher-order symmetries

We have shown how the six non-autonomous DΔEs are derived from the DL scheme, by associating each equation with a certain linear integral equation in the form of (2.6). From the viewpoint of the DL approach, this guarantees the integrability of the resulting nonlinear DΔEs, as long as their respective linear integral equations are solvable. Here by integrability we mean the existence of an exact solution possessing an infinite number of free parameters (e.g. soliton-type solution), while the initial boundary value problem is not discussed. Below we present soliton solutions of these DΔEs, by performing a very special reduction on the measure in the respective linear integral equations.

We start with the formal tau functions (3.7), (4.9) and (5.11) in order to construct soliton-type solutions, instead of the linear integral equation (2.6), since the tau functions defined as such already contain the key ingredients of the linear integral equation, namely the integration domain and measure, the plane wave factors as well as the Cauchy kernel. The procedure is exactly the same as that given in [8]. Hence, we only provide a short guide and directly list the results below. For equations (1.4a), (1.4b) and (1.4c), we take the measure

$$\mathrm{d}\zeta (k,k^{\prime })=\sum _{i=1}^N\sum _{j=1}^{N^{\prime }}\frac{A_{i,j}}{{(2\pi \mathfrak{i})}^2}\frac{1}{k-k_i}\frac{1}{k^{\prime }-k_j^{\prime }}\mathrm{d}k\mathrm{d}{k}^{\prime }\text{with}\mathfrak{i}\text{being the imaginary unit},$$ 6.1

which results in the determinant solution (i.e. soliton solution)

$$\tau =det\hspace{0.17em}(1+\mathit{A}\mathit{M}),$$ 6.2

where 1 denotes the N × N unit matrix, A = (A_i,j)_N×N′ is an arbitrary matrix and the entries of M = (M_j,i)_N′×N are given by

$$M_{j,i}=\frac{{\rho }_{k_i}{\sigma }_{k_j^{\prime }}}{k_i+k_j^{\prime }},\text{with}\rho \text{and}\sigma \text{given by\hspace{0.17em}}(3.1).$$

Here, N and N′ are arbitrary positive integers. Similarly, for equations (1.5a) and (1.5b), we take a special measure

$$\mathrm{d}\zeta (k,k^{\prime })=\sum _{i,j=1}^{2N}\frac{A_{i,j}}{{(2\pi \mathfrak{i})}^2}\frac{1}{k-k_i}\frac{1}{k^{\prime }-k_j^{\prime }}\mathrm{d}k\mathrm{d}{k}^{\prime },A_{i,j}=-A_{j,i},$$ 6.3

which obeys the antisymmetry condition in (4.7). In this case, the tau function is determined by

$${\tau }^2=det(1+\mathit{A}\mathit{M}),$$ 6.4

where A = (A_i,j)_2N×2N is an antisymmetric matrix and the Cauchy matrix M = (M_j,i)_2N×2N has its (j, i)-entry

$$M_{j,i}={\rho }_{k_i}\frac{1}{2}\frac{k_i-k_j^{\prime }}{k_i+k_j^{\prime }}{\sigma }_{k_j^{\prime }},\text{in which}\rho \text{and}\sigma \text{are defined by\hspace{0.17em}}(4.1).$$

For the CKP class, we consider the following measure reduction:

$$\mathrm{d}\zeta (k,k^{\prime })=\sum _{i,j=1}^N\frac{A_{i,j}}{{(2\pi \mathfrak{i})}^2}\frac{1}{k-k_i}\frac{1}{k^{\prime }-k_j^{\prime }}\mathrm{d}k\mathrm{d}{k}^{\prime },A_{i,j}=A_{j,i},$$ 6.5

where we have respected the condition in (5.7). Then the formal tau function (5.11) turns out to be

$$\tau =det(1+\mathit{A}\mathit{M}),$$ 6.6

in which A = (A_i,j)_N×N is a symmetric matrix and the (i, j)-entry of M = (M_j,i)_N,N is defined as

$$M_{j,i}=\frac{{\rho }_{k_i}{\sigma }_{k_j^{\prime }}}{k_i+k_j^{\prime }},\text{where}\rho \text{and}\sigma \text{are defined as\hspace{0.17em}}(5.1).$$

The DL also provides a perspective to understand the MDC property of the six DΔEs, from their respective solution structures. We focus on the formal tau functions (3.7), (4.9) and (5.11), since the soliton-type expressions (6.2), (6.4) and (6.6) are the natural consequences arising from the measure reductions (6.1), (6.3) and (6.5), respectively. We take equation (1.5a) as an example. The formal solution (4.9) relies on its flow variables p_i, p_j, n_i and n_j through the infinite matrix C, whose dynamics are completely determined by the plane wave factor ρ_kσ_k′ given in (4.1). Observing that all independent variables are on the same footing² in (4.1), we conclude that DΔEs in the form of (1.5a) with regard to different lattice parameters and variables share the same non-trivial solution (4.9); in other words, these equations are compatible with each other. This is the MDC property that we normally adopt in the theory of discrete integrable systems, and here it is applied to the differential-difference case.

Furthermore, the derivation of (1.4a), (1.4b) and (1.4c) shows that the three equations have the same tau function (3.7) as the common solution, and such a tau function simultaneously (cf. [8]) serves as a solution of (1.1). This implies that these four equations are multi-dimensionally consistent, within the DL scheme; in other words, equations (1.4a), (1.4b), (1.4c) and (1.1) should be treated as compatible flows in the AKP class. Similarly, equations (1.5a), (1.5b) and (1.2) are also multi-dimensionally consistent with each other, while equation (1.6) is compatible with (1.3), from the same viewpoint. This is also the MDC property, namely equations in different forms can still be consistent with each other. As a remark, we would like to note that here the MDC is only verified on the solutions generated by the DL framework. The verification of the consistency on the level of equations still remains as future work. We believe the assertion is correct, as the solutions following from the DL already possess an infinite number of degrees of freedom.

With the help of the solution structures of the AKP, BKP and CKP classes discussed above, we are now allowed to explain how the higher-order symmetries in the continuous KP-type hierarchies are generated from the lattice parameters. The idea to realize this is to compare the effective plane wave factors in each discrete KP class with their corresponding continuous analogues (which are given in [35]). In the AKP class, we require

$$\begin{array}{rl}{\rho }_k{\sigma }_{k^{\prime }}& =\prod _{i=1}^{\mathrm{\infty }}{\left(\frac{p_i+k}{p_i-k^{\prime }}\right)}^{n_i}=\exp \{\sum _{j=1}^{\mathrm{\infty }}(k^j-{(-k^{\prime })}^j)\frac{{(-1)}^{j-1}}{j}\sum _{i=1}^{\mathrm{\infty }}\frac{n_i}{p_i^j}\}\\ \hspace{0.17em}& =\exp \{\sum _{j=1}^{\mathrm{\infty }}(k^j-{(-k^{\prime })}^j)x_j\}.\end{array}$$

Therefore, the relationship between the continuous variables x_j and the lattice parameters p_i is given by

$$x_j=\frac{{(-1)}^{j-1}}{j}\sum _{i=1}^{\mathrm{\infty }}\frac{n_i}{p_i^j}\text{for}j=1,2,\dots ,\text{which results in}{\mathrm{\partial }}_{p_i}=\sum _{j=1}^{\mathrm{\infty }}{(-1)}^j\frac{n_i}{p_i^{j+1}}{\mathrm{\partial }}_{x_j}.$$ 6.7

Likewise, as the form of the plane wave factor is different, we require in the BKP and CKP classes

$$\begin{array}{rl}{\rho }_k{\sigma }_{k^{\prime }}& =\prod _{i=1}^{\mathrm{\infty }}{\left(\frac{p_i+k}{p_i-k}\frac{p_i+k^{\prime }}{p_i-k^{\prime }}\right)}^{n_i}=\exp \{\sum _{j=0}^{\mathrm{\infty }}(k^{2j+1}+k^{\prime 2j+1})\frac{2}{2j+1}\sum _{i=1}^{\mathrm{\infty }}\frac{n_i}{p_i^{2j+1}}\}\\ \hspace{0.17em}& =\exp \{\sum _{j=0}^{\mathrm{\infty }}(k^{2j+1}+k^{\prime 2j+1})x_{2j+1}\},\end{array}$$

and subsequently we obtain

$$x_{2j+1}=\frac{2}{2j+1}\sum _{i=1}^{\mathrm{\infty }}\frac{n_i}{p_i^{2j+1}}\text{for}j=0,1,2,\dots ,\text{which leads to}{\mathrm{\partial }}_{p_i}=-2\sum _{j=0}^{\mathrm{\infty }}\frac{n_i}{p_i^{2j+2}}{\mathrm{\partial }}_{x_{2j+1}}.$$ 6.8

The transformations between the derivatives of lattice parameters p_i and continuous flow variables x_j listed in (6.7) and (6.8) provide a way of generating all the higher-order symmetries of the continuous KP-type from the lattice parameters. They are the counterparts of the result in the generating PDE of the discrete KdV equation given in [19] on the KP level, although the generating PDEs for the discrete KP-type equations are not yet clear so far.

7. Conclusion

Based on the DL framework for the discrete AKP, BKP and CKP equations, we introduce a new perspective to construct integrable nonlinear equations, by thinking of the lattice parameters as the independent variables and treating them together with the discrete variables equally. As a result, six novel non-autonomous differential equations are proposed, including two (2 + 2)-dimensional non-autonomous DΔEs, namely (1.5a) and (1.6) from BKP and CKP classes, respectively, as well as four (2 + 1)-dimensional non-autonomous semi-discrete equations, i.e. (1.4a), (1.4b) and (1.4c) from the AKP class and (1.5b) from the BKP class.

It was already shown that non-autonomous DΔEs often play the role of master symmetries for lower-dimensional discrete integrable systems [36], while the master symmetry theory for higher-dimensional lattice equations is not yet clear. These new equations potentially provide us with an insight into understanding master symmetries of three-dimensional integrable discrete models. Furthermore, it seems that (2 + 2)-dimensional non-autonomous integrable models of Toda-type, such as equations (1.5a) and (1.6), have never appeared in the literature so far, to the best of the authors’ knowledge. Their geometric interpretation still needs to be discovered.

In addition to symmetries and master symmetries, we also note that the integrability of discrete KP-type equations was also proven from the perspective of conservations laws. In [37], conservation laws for the discrete AKP and BKP equations were constructed, while it remains a problem for the discrete CKP equation.

The ultimate goal is to find closed-form generating PDEs for higher-dimensional integrable hierarchies and their integrability structures. This also remains an open problem for future study.

Data accessibility

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Authors' contributions

Both authors conceived the idea and established the direct linearization scheme together. F.W.N. provided equation (1.5a) as the first example, and then the authors together found the three equations in (1.4), while (1.5b) and (1.6) were obtained by W.F. The manuscript was drafted by W.F. and was revised by F.W.N. Both authors gave final approval for publication and agree to be held accountable for the work performed therein.

Competing interests

We declare we have no competing interests.

Funding

W.F. was sponsored by the National Natural Science Foundation of China (grant nos 11901198 and 11871396) and by Shanghai Pujiang Program (grant no. 19PJ1403200). This project was also partially supported by the Science and Technology Commission of Shanghai Municipality (grant no. 18dz2271000).