A systematic construction of integrable delay-difference and delay-differential analogues of soliton equations

Delay-differential equations have been used as mathematical models in various fields of science and engineering such as traffic flow, population dynamics, nonlinear optics, fluid mechanics and infectious disease [1, 2]. Because of their importance in various fields, the mathematical properties of delay differential equations have been actively studied. For example, exact solutions of delay differential equations play an important role in the study of traffic flow [3, 4, 5, 6].

In recent years, there has been active research on integrability of delay differential equations. Quispel et al obtained a delay-differential equation which has a continuum limit to the first Painlevé equation [7]. This delay-differential equation was derived by a similarity reduction of the Lotka-Volterra (LV) equation, which is an integrable differential-difference equation. Levi and Winternitz also obtained delay-differential equations as reductions based on the symmetry group of the two-dimensional Toda lattice (2DTL) [8]. In addition, Grammaticos et al [9, 10] introduced delay-differential Painlevé equations by the method blending the Painlevé analysis with the singularity confinement. After their monumental works, several important studies have revealed mathematical properties of integrable (ordinary) delay-differential equations [11, 12, 13, 14, 15, 16, 17].

On the other hand, there are few examples of integrable partial delay-differential equations (the word “partial” means including multi independent variables). Villarroel and Ablowitz found the Lax pair for a delay-differential analogue of the 2DTL equation and established the inverse scattering formalism [18]. Recently, we constructed the $N$-soliton solution of the delay-differential analogues of the 2DTL equation [19]. Although there have been such works, research on integrable partial delay-differential equations is not fully developed.

In this paper, we propose a systematic method for constructing integrable delay-difference and delay-differential analogues of soliton equations and their multi-soliton solutions. Our construction starts from the discrete KP equation (or discrete 2DTL equation) and uses reduction and delay-differential limit. As examples of our method, we present delay-difference and delay-differential analogues of the LV, Toda lattice (TL), and sine-Gordon (sG) equations and their exact $N$-soliton solutions.

This paper is organized as follows. In the rest of this section, we show a systematic method to construct integrable delay-difference and delay-differential analogues of soliton equations. In sections 2, 3, and 4, we construct delay-difference and delay-differential analogues of the LV, TL, and sG equations and their $N$-soliton solutions. Section 5 is devoted to conclusions.

In this section, we show the construction of the delay-difference and delay-differential analogue of the LV equation and their $N$-soliton solutions. We show the detail of our method through this section.

First, we consider the discrete KP equation (1.1) and its $N$-soliton solution in the Gram determinant form [23]:

	$\displaystyle f_{n,m,k}=\det\left(\delta_{ij}+\frac{\phi_{i}\psi_{j}}{p_{i}-q_{j}}\right)_{1\leq i,j\leq N}=\frac{\prod_{i=1}^{N}\phi_{i}}{\prod_{j=1}^{N}\phi_{j}}\det\left(\delta_{ij}\frac{\phi_{j}}{\phi_{i}}+\frac{\phi_{j}\psi_{j}}{p_{i}-q_{j}}\right)_{1\leq i,j\leq N}$		(4)
	$\displaystyle\hskip 28.45274pt=\det\left(\delta_{ij}+\frac{\phi_{j}\psi_{j}}{p_{i}-q_{j}}\right)_{1\leq i,j\leq N}\,,$
	$\displaystyle\phi_{i}(n,m,k)=\beta_{i}(1-ap_{i})^{-n}(1-bp_{i})^{-m}(1-cp_{i})^{-k}\,,$
	$\displaystyle\psi_{i}(n,m,k)=\gamma_{i}(1-aq_{i})^{n}(1-bq_{i})^{m}(1-cq_{i})^{k}\,.$

This Gram determinant form of the $N$-soliton solution can be rewritten as

	$\displaystyle f_{n,m,k}=\sum_{I\subset\{1,\ldots,N\}}\prod_{i\in I}\Phi_{i}\prod_{i<j,\ i,j\in I}\frac{(p_{i}-p_{j})(q_{i}-q_{j})}{(p_{i}-q_{j})(q_{i}-p_{j})}\,,$		(5)
	$\displaystyle\Phi_{i}(n,m,k)=\beta_{i}\gamma_{i}\left(\frac{1-aq_{i}}{1-ap_{i}}\right)^{n}\left(\frac{1-bq_{i}}{1-bp_{i}}\right)^{m}\left(\frac{1-cq_{i}}{1-cp_{i}}\right)^{k}\,.$

Here $\displaystyle\sum_{I\subset\{1,\ldots,N\}}$ is taken to be summed over all possible combinations in the set $\{1,\ldots,N\}$. Now, we apply the following reduction condition to the discrete KP equation (1.1):

$$f_{n,m,k+1}=f_{n+2,m+\alpha,k}\,,$$

(6)

where the free parameter $\alpha$ is a fixed real value considered as the delay. Setting $c=-a$ and $\delta=2b/(a-b)$, we obtain the following discrete bilinear equation by the reduction of the discrete KP equation (1.1):

$$(1+\delta)f_{n}^{m+1+\alpha}f_{n-1}^{m}-\delta f_{n+1}^{m+\alpha}f_{n-2}^{m+1}-f_{n}^{m+\alpha}f_{n-1}^{m+1}=0\,,$$

(7)

where $f_{n}^{m}\equiv f_{n,m,k}$. More precisely, applying the reduction condition (6) to equation (1.1), we have made the index $k+1$ of $f$ become $k$. Then we can omit the variable $k$ since the iterations of $k$ vanish. We can rewrite the bilinear equation (7) as follows by using Hirota’s D-operators:

	$\displaystyle\left(2\sinh\left(\frac{D_{m}}{2}\right)\sinh\left(\frac{D_{n}}{2}+\frac{\alpha D_{m}}{2}\right)-2\delta\sinh\left(\frac{D_{n}-D_{m}}{2}\right)\sinh\left(D_{n}+\frac{\alpha D_{m}}{2}\right)\right)f_{n}^{m}\cdot f_{n}^{m}$
	$\displaystyle=0\,.$		(8)

Here Hirota’s D-operators are defined by

$$D^{l}_{t}{g(t)}\cdot{h(t)}=\left(\frac{\partial}{\partial t}-\frac{\partial}{\partial s}\right)^{l}g(t)h(s)\Bigr{|}_{s=t}\,,\qquad e^{D_{m}}g_{m}\cdot h_{m}=g_{m+1}h_{m-1}\,.$$

(9)

Next we consider the reduction of the $N$-soliton solution. By applying the constraint

$$\left(\frac{1-aq_{i}}{1-ap_{i}}\right)^{2}\left(\frac{1-bq_{i}}{1-bp_{i}}\right)^{\alpha}=\frac{1-cq_{i}}{1-cp_{i}}$$

(10)

to the $N$-soliton solution (4) and (5), we obtain

	$\displaystyle\Phi_{i}(n,m,k+1)$	$\displaystyle=$	$\displaystyle\beta_{i}\gamma_{i}\left(\frac{1-aq_{i}}{1-ap_{i}}\right)^{n}\left(\frac{1-bq_{i}}{1-bp_{i}}\right)^{m}\left(\frac{1-cq_{i}}{1-cp_{i}}\right)^{k+1}$		(11)
		$\displaystyle=$	$\displaystyle\beta_{i}\gamma_{i}\left(\frac{1-aq_{i}}{1-ap_{i}}\right)^{n+2}\left(\frac{1-bq_{i}}{1-bp_{i}}\right)^{m+\alpha}\left(\frac{1-cq_{i}}{1-cp_{i}}\right)^{k}$
		$\displaystyle=$	$\displaystyle\Phi_{i}(n+2,m+\alpha,k)\,,$

thus the reduction condition (6) is satisfied. Setting $c=-a$, $\delta=2b/(a-b)$, $k=0$, and replacing $p_{i}$ and $q_{i}$ by $(-2p_{i}-1)/a$ and $(-2q_{i}-1)/a$ respectively, we obtain the $N$-soliton solution of the bilinear equation (7) by the reduction to (4) and (5):

	$\displaystyle f_{n}^{m}=\det\left(\delta_{ij}+\frac{\Phi_{j}}{p_{i}-q_{j}}\right)_{1\leq i,j\leq N}=\sum_{I\subset\{1,\ldots,N\}}\prod_{i\in I}\Phi_{i}\prod_{i<j,\ i,j\in I}\frac{(p_{i}-p_{j})(q_{i}-q_{j})}{(p_{i}-q_{j})(q_{i}-p_{j})}\,,$		(12)
	$\displaystyle\Phi_{i}(n,m)=\beta_{i}\gamma_{i}\left(\frac{1+q_{i}}{1+p_{i}}\right)^{n}\left(\frac{1+\delta+\delta q_{i}}{1+\delta+\delta p_{i}}\right)^{m}\,,$
	$\displaystyle\frac{q_{i}}{p_{i}}=\left(\frac{1+q_{i}}{1+p_{i}}\right)^{2}\left(\frac{1+\delta+\delta q_{i}}{1+\delta+\delta p_{i}}\right)^{\alpha}\,.$

The delay parameter $\alpha$ appears in the bilinear equation (7) as shifts of the discrete time variable $m$. We remark that the bilinear equation (7) and the $N$-soliton solution (12) in the case of $\alpha=0$ (thus $p_{i}=1/q_{i}$) are known as the bilinear equation of the discrete LV equation and its $N$-soliton solution [24, 25]. Thus equation (7) can be seen as the bilinear equation of the delay-difference analogue of the LV equation.

By the dependent variable transformation

$$u_{n}^{m}=\frac{f_{n+1}^{m+\alpha}f_{n-2}^{m+1}}{f_{n}^{m+\alpha}f_{n-1}^{m+1}}\,,$$

(13)

the bilinear equation (7) is transformed into the nonlinear delay-difference equation

$$\frac{u_{n}^{m+1+\alpha}u_{n-1}^{m}}{u_{n}^{m+\alpha}u_{n-1}^{m+1}}=\frac{(1+\delta u_{n+1}^{m+\alpha})(1+\delta u_{n-2}^{m+1})}{(1+\delta u_{n}^{m+\alpha})(1+\delta u_{n-1}^{m+1})}\,,$$

(14)

which is the delay-difference analogue of the LV equation. If $\alpha=0$, equation (14) is just a division of the discrete LV equation [24]:

$$\frac{u_{n}^{m+1}}{u_{n}^{m}}=\frac{1+\delta u_{n+1}^{m}}{1+\delta u_{n-1}^{m+1}}\,,\qquad\frac{u_{n-1}^{m+1}}{u_{n-1}^{m}}=\frac{1+\delta u_{n}^{m}}{1+\delta u_{n-2}^{m+1}}\,.$$

(15)

Now, we apply the delay-differential limit

$$\delta\to 0\,,\qquad m\delta=t\,,\qquad\alpha\delta=2\tau\,,$$

(16)

where $\tau$ is a constant value called the delay parameter. The delay-differential limits of (7) and (12) are calculated respectively as follows:

$\displaystyle D_{t}{f_{n}(t+\tau)}\cdot{f_{n-1}(t-\tau)}-f_{n+1}(t+\tau)f_{n-2}(t-\tau)+f_{n}(t+\tau)f_{n-1}(t-\tau)=0\,,$

(17)

	$\displaystyle f_{n}(t)=\det\left(\delta_{ij}+\frac{\Phi_{j}}{p_{i}-q_{j}}\right)_{1\leq i,j\leq N}=\sum_{I\subset\{1,\ldots,N\}}\prod_{i\in I}\Phi_{i}\prod_{i<j,\ i,j\in I}\frac{(p_{i}-p_{j})(q_{i}-q_{j})}{(p_{i}-q_{j})(q_{i}-p_{j})}\,,$		(18)
	$\displaystyle\Phi_{i}(n,t)=\beta_{i}\gamma_{i}\left(\frac{1+q_{i}}{1+p_{i}}\right)^{n}e^{(q_{i}-p_{i})t}\,,$
	$\displaystyle\frac{q_{i}}{p_{i}}=\left(\frac{1+q_{i}}{1+p_{i}}\right)^{2}e^{2\tau(q_{i}-p_{i})}\,.$

Here the bilinear equation (17) can be rewritten as

$\displaystyle\left(D_{t}\sinh\left(\frac{D_{n}}{2}+\tau D_{t}\right)-2\sinh\left(\frac{D_{n}}{2}\right)\sinh\left(D_{n}+\tau D_{t}\right)\right)f_{n}(t)\cdot f_{n}(t)=0\,.$

(19)

The bilinear equation (17) is a delay differential-difference equation which includes the delay $\tau$ as shifts of the continuous time variable $t$. Putting $\tau=0$ (thus $p_{i}=1/q_{i}$), we can easily check that equation (17) becomes the bilinear equation of the LV equation and (18) becomes the $N$-soliton solution of the LV equation [26]. Thus we claim that equation (17) is the bilinear equation of the integrable delay LV equation and the bilinear equation (7) is the fully discrete analogue of (17).

Let us move the discussion to the nonlinear form of the delay LV equation. Via the dependent variable transformation

$$u_{n}(t)=\frac{f_{n+1}(t+\tau)f_{n-2}(t-\tau)}{f_{n}(t+\tau)f_{n-1}(t-\tau)}\,,$$

(25)

the bilinear equation (17) is transformed into the nonlinear delay-differential equation

$$\frac{\mathrm{d}}{\mathrm{d}t}\log\frac{u_{n}(t+\tau)}{u_{n-1}(t-\tau)}=u_{n+1}(t+\tau)-u_{n}(t+\tau)-u_{n-1}(t-\tau)+u_{n-2}(t-\tau)\,,$$

(26)

which is the nonlinear form of (17). If $\tau=0$, equation (26) is just a subtraction of the following nonlinear forms of the LV equation [26]:

$$\frac{\mathrm{d}}{\mathrm{d}t}\log u_{n}(t)=u_{n+1}(t)-u_{n-1}(t)\,,\qquad\frac{\mathrm{d}}{\mathrm{d}t}\log u_{n-1}(t)=u_{n}(t)-u_{n-2}(t)\,.$$

(27)

Next, we derive the bilinear form of the delay LV equation (17) and the $N$-soliton solution (18) directly from the semi-discrete KP equation [28]

$$acD_{t}f_{n+1}^{k}(t)\cdot f_{n}^{k+1}(t)-(a-c)(f_{n+1}^{k+1}(t)f_{n}^{k}(t)-f_{n+1}^{k}(t)f_{n}^{k+1}(t))=0\,.$$

(28)

We remark that the bilinear equation (28) and its solutions are derived by the continuum limit $b\to 0$, $mb=t$ of the discrete KP equation (1.1) and its solutions.

The Wronskian solution of the semi-discrete KP equation (28) is given as follows [28]:

	$\displaystyle f_{n}^{k}(t)=\left|\begin{array}[]{cccc}\phi_{1}(n,k,t)&\phi_{1}^{(1)}(n,k,t)&\cdots&\phi_{1}^{(N-1)}(n,k,t)\\ \phi_{2}(n,k,t)&\phi_{2}^{(1)}(n,k,t)&\cdots&\phi_{2}^{(N-1)}(n,k,t)\\ \vdots&\vdots&\cdots&\vdots\\ \phi_{N}(n,k,t)&\phi_{N}^{(1)}(n,k,t)&\cdots&\phi_{N}^{(N-1)}(n,k,t)\end{array}\right|\,,$		(33)
	$\displaystyle\frac{\phi_{i}(n,k,t)-\phi_{i}(n-1,k,t)}{a}=\frac{\partial\phi_{i}(n,k,t)}{\partial t}\,,\quad\frac{\phi_{i}(n,k,t)-\phi_{i}(n,k-1,t)}{c}=\frac{\partial\phi_{i}(n,k,t)}{\partial t}\,,$
	$\displaystyle\phi_{i}^{(l)}(n,k,t)\equiv\frac{\partial^{l}\phi_{i}(n,k,t)}{\partial t^{l}}\,.$

The $N$-soliton solution of the semi-discrete KP equation is given by choosing the elements in the Wronskian solution as

	$\displaystyle\phi_{i}(n,k,t)=(1-ap_{i})^{-n}(1-cp_{i})^{-k}e^{p_{i}t+\zeta_{0i}}+(1-aq_{i})^{-n}(1-cq_{i})^{-k}e^{q_{i}t+\eta_{0i}}\,.$		(34)
	$\displaystyle(i=1,2,\ldots,N)$

Now, we can obtain the bilinear equation (17) by applying the reduction condition

$$f_{n}^{k+1}(t)\Bumpeq f_{n+2}^{k}(t+2\tau)$$

(35)

and setting $a=2$, $c=-2$ to the semi-discrete KP equation (28). Here, the relation $g_{n}^{k}(t)\Bumpeq h_{n}^{k}(t)$ is defined by

$$g_{n}^{k}(t)=\left(C_{0}C_{1}^{n}C_{2}^{k}e^{C_{3}t}\right)h_{n}^{k}(t)\,,\quad C_{l}=\mathrm{const}.\,.\quad(l=0,1,2,3)$$

(36)

To realize this reduction condition ($C_{1}=C_{2}=1,C_{3}=0$) for the $N$-soliton solutions, we can apply the constraint

$$(1-ap_{i})^{-2}(1-cp_{i})e^{2\tau p_{i}}=(1-aq_{i})^{-2}(1-cq_{i})e^{2\tau q_{i}}$$

(37)

to (34). Replacing $p_{i}$ and $q_{i}$ by $(-2p_{i}-1)/a$ and $(-2q_{i}-1)/a$ respectively and setting $a=2$, $c=-2$, we obtain the $N$-soliton solution of the delay LV equation in the Wronskian form

	$\displaystyle f_{n}(t)=\left|\begin{array}[]{cccc}\phi_{1}(n,t)&\phi_{1}^{(1)}(n,t)&\cdots&\phi_{1}^{(N-1)}(n,t)\\ \phi_{2}(n,t)&\phi_{2}^{(1)}(n,t)&\cdots&\phi_{2}^{(N-1)}(n,t)\\ \vdots&\vdots&\cdots&\vdots\\ \phi_{N}(n,t)&\phi_{N}^{(1)}(n,t)&\cdots&\phi_{N}^{(N-1)}(n,t)\end{array}\right|\,,$		(42)
	$\displaystyle\phi_{i}(n,t)=(1+p_{i})^{-n}e^{-p_{i}t+\tilde{\zeta}_{0i}}+(1+q_{i})^{-n}e^{-q_{i}t+\tilde{\eta}_{0i}}\,,\qquad(i=1,2,\ldots,N)$
	$\displaystyle\frac{q_{i}}{p_{i}}=\left(\frac{1+q_{i}}{1+p_{i}}\right)^{2}e^{2\tau(q_{i}-p_{i})}\,.$

In this section, we construct a delay differential-difference equation that should be called an integrable delay TL equation by using our method.

We first apply the reduction condition

$$f_{n,m,k+1}=f_{n+1,m+1+\alpha,k}$$

(50)

to the discrete KP equation (1.1) and its $N$-soliton solution (4). The independent variables $n$ and $m$ are considered to be the discrete space variable and discrete time variable respectively, and the parameter $\alpha$ is a delay parameter. Replacing $p_{i}$ and $q_{i}$ by $(1-p_{i})/c$ and $(1-q_{i})/c$ respectively and setting $\alpha\delta=(a-c)/a$, $\delta=b/(b-c)$, we have

$$f_{n}^{m+1+\alpha}f_{n}^{m-1}-\alpha\delta^{2}f_{n+1}^{m+\alpha}f_{n-1}^{m}-(1-\alpha\delta^{2})f_{n}^{m+\alpha}f_{n}^{m}=0\,,$$

(51)

	$\displaystyle f_{n}^{m}$	$\displaystyle=$	$\displaystyle\det\left(\delta_{ij}+\frac{\Phi_{j}}{p_{i}-q_{j}}\right)_{1\leq i,j\leq N}=\sum_{I\subset\{1,\ldots,N\}}\prod_{i\in I}\Phi_{i}\prod_{i<j,\ i,j\in I}\frac{(p_{i}-p_{j})(q_{i}-q_{j})}{(p_{i}-q_{j})(q_{i}-p_{j})}\,,$		(52)
	$\displaystyle\Phi_{i}$	$\displaystyle=$	$\displaystyle\beta_{i}\gamma_{i}\left(\frac{q_{i}-\alpha\delta}{p_{i}-\alpha\delta}\right)^{n}\left(\frac{1-\delta q_{i}}{1-\delta p_{i}}\right)^{m}\,,$
	$\displaystyle\frac{q_{i}}{p_{i}}$	$\displaystyle=$	$\displaystyle\frac{q_{i}-\alpha\delta}{p_{i}-\alpha\delta}\left(\frac{1-\delta q_{i}}{1-\delta p_{i}}\right)^{1+\alpha}\,.$

The bilinear equation (51) is rewritten as

	$\displaystyle\left(2\sinh\left(\frac{D_{m}}{2}\right)\sinh\left(\frac{D_{m}}{2}+\frac{\alpha D_{m}}{2}\right)-2\alpha\delta^{2}\sinh\left(\frac{D_{n}}{2}\right)\sinh\left(\frac{D_{n}}{2}+\frac{\alpha D_{m}}{2}\right)\right)f_{n}^{m}\cdot f_{n}^{m}$
	$\displaystyle=0\,.$		(53)

We can consider that equation (51) is the bilinear equation of the delay-difference analogue of the TL equation. We call this the delay discrete TL equation.

By the dependent variable transformation

$$1+V_{n}^{m}=\frac{f_{n+1}^{m+\alpha}f_{n-1}^{m}}{f_{n}^{m+\alpha}f_{n}^{m}}\,,$$

(54)

we can transform (51) into the nonlinear delay-difference equation

$$\frac{(1+V_{n}^{m+1+\alpha})(1+V_{n}^{m-1})}{(1+V_{n}^{m+\alpha})(1+V_{n}^{m})}=\frac{(1+\alpha\delta^{2}V_{n+1}^{m+\alpha})(1+\alpha\delta^{2}V_{n-1}^{m})}{(1+\alpha\delta^{2}V_{n}^{m+\alpha})(1+\alpha\delta^{2}V_{n}^{m})}\,,$$

(55)

which is the delay discrete TL equation.

Now, we apply the delay-differential limit

$$\delta\to 0\,,\qquad m\delta=t\,,\qquad\alpha\delta=2\tau$$

(56)

to (51) and (52), where $t$ is the continuous time variable and $\tau$ is the delay parameter. Then we obtain

$\displaystyle D_{t}{f_{n}(t+\tau)}\cdot{f_{n}(t-\tau)}-2\tau(f_{n+1}(t+\tau)f_{n-1}(t-\tau)-f_{n}(t+\tau)f_{n}(t-\tau))=0$

(57)

and

	$\displaystyle f_{n}(t)$	$\displaystyle=$	$\displaystyle\det\left(\delta_{ij}+\frac{\Phi_{j}}{p_{i}-q_{j}}\right)_{1\leq i,j\leq N}=\sum_{I\subset\{1,\ldots,N\}}\prod_{i\in I}\Phi_{i}\prod_{i<j,\ i,j\in I}\frac{(p_{i}-p_{j})(q_{i}-q_{j})}{(p_{i}-q_{j})(q_{i}-p_{j})}\,,$		(58)
	$\displaystyle\Phi_{i}$	$\displaystyle=$	$\displaystyle\beta_{i}\gamma_{i}\left(\frac{q_{i}-2\tau}{p_{i}-2\tau}\right)^{n}e^{(p_{i}-q_{i})t}\,,$
	$\displaystyle\frac{q_{i}}{p_{i}}$	$\displaystyle=$	$\displaystyle\frac{q_{i}-2\tau}{p_{i}-2\tau}e^{2\tau(p_{i}-q_{i})}\,.$

Here the bilinear equation (57) is equivalent to

$$\left(D_{t}\sinh\left(\tau D_{t}\right)-4\tau\sinh\left(\frac{D_{n}}{2}\right)\sinh\left(\frac{D_{n}}{2}+\tau D_{t}\right)\right)f_{n}(t)\cdot f_{n}(t)=0\,.$$

(59)

Calculating the limit of this equation as $\tau\to 0$, we obtain the bilinear TL equation [30]

$$\left(D^{2}_{t}-4\sinh^{2}\left(\frac{D_{n}}{2}\right)\right)f_{n}(t)\cdot f_{n}(t)=0\,.$$

(60)

The last relation of (58) is described by $(p_{i}+1/p_{i}-q_{i}-1/q_{i})\tau+O(\tau^{2})=0$, thus we have $p_{i}=1/q_{i}$ as $\tau\to 0$. Therefore, we can check that the limit of (58) as $\tau\to 0$ is the $N$-soliton solution of the TL equation [27, 31, 30]. Thus we claim that equation (57) is the bilinear equation of the delay TL equation.

We present the nonlinear form of the bilinear equation of the delay TL equation (57) under the dependent variable transformation

$$1+V_{n}(t)=\frac{f_{n+1}(t+\tau)f_{n-1}(t-\tau)}{f_{n}(t+\tau)f_{n}(t-\tau)}\,.$$

(61)

By using this transformation, we can transform (57) into the delay-differential equation

	$\displaystyle\left(\frac{1}{2\tau}\right)\frac{\mathrm{d}}{\mathrm{d}t}\log\frac{1+V_{n}(t+\tau)}{1+V_{n}(t-\tau)}$
	$\displaystyle=V_{n+1}(t+\tau)+V_{n-1}(t-\tau)-V_{n}(t+\tau)-V_{n}(t-\tau)\,,$		(62)

which is the nonlinear form of (57). The limit of (3) as $\tau\to 0$ is the nonlinear form of the TL equation [27, 31, 30]:

$$\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\log(1+V_{n}(t))=V_{n+1}(t)-2V_{n}(t)+V_{n-1}(t)\,.$$

(63)