Miura-Type Transformations for Integrable Lattices in 3D

Abstract

This article studies a class of integrable semi-discrete equations with one continuous and two discrete independent variables. At present, in the literature there are nine integrable equations of the form $u_{n+1,x}^j=f(u_{n,x}^j,u_n^{j+1},u_n^j,u_{n+1}^j,u_{n+1}^{j-1})$ up to point transformations. An efficient method based on some relation that generalizes the notion of the local conservation law is proposed for searching for Miura-type transformations relating to semi-discrete equations in 3D. The efficiency of the method is illustrated with the equations from the list. For one of the equations, which is little studied, the continuum limit is calculated. For this equation, the problem of finite-field reductions in the form of Darboux integrable systems of equations of a hyperbolic type is discussed. For reductions of small orders, $N=1$ and $N=2$, complete sets of characteristic integrals are presented. Note that the existence of characteristic integrals makes it possible to construct particular solutions to the original lattice. For the case $N=1$, an explicit solution was found in this paper. A new semi-discrete equation is found that lies beyond the considered class. For this equation, the Lax pair is presented.

1. Introduction

Nonlinear integrable differential-difference equations with one continuous and two discrete variables are intermediate objects between the class of integrable nonlinear equations of dimension 1 + 2, such as the Toda lattice, the Kadomtsev–Petviashvili equation, the Davey–Stewartson equation, etc. (see [1,2,3]), and the class of fundamental discrete Hirota–Miwa-type equations (see [4,5,6,7]). They also have applications in the theory of Laplace transformations in the class of linear differential-difference equations of a hyperbolic type [8]. More precisely, the sequence of Laplace invariants satisfies some integrable differential-difference equations of the form (1), related by a point transformation to Equation (L7) from the list below. Such lattices can find application in studying the nonlinear dynamics of localized magnetic inhomogeneities in such magnetically ordered substances as ferromagnets, antiferromagnets with weak ferromagnetism, and magnetoelastic and magnetoelectric interactions [9,10]. Equations of the type under consideration have potential applications in the problem of describing dislocations in media with a microstructure, as well as in the nonlinear theory of elastic and nonelastic deformations accompanied with the deep reconstruction of an initially ideal lattice: the switching of interatomic bonds, the changing of the class of symmetry, the formation of new phases, singular defects and heterogeneities, and the fragmentation of the lattice (see, for instance, [11,12]). Lattices of the type under consideration can be used in the derivation of difference schemes for finding numerical solutions to nonlinear partial differential equations (see [13,14]).

Consider the class of integrable differential-difference equations with one continuous and two discrete independent variables of the following form:

$$u_{n+1,x}^j=f(u_{n,x}^j,u_n^{j+1},u_n^j,u_{n+1}^j,u_{n+1}^{j-1}),$$

(1)

where the unknown function $u=u_n^j\left(x\right)$ depends on the continuous x and the discrete n and j. The subscript x denotes the derivative $u_x=\frac{d}{dx}u$. Below, we give a list of known integrable models of the form (1), without pretending to be complete. The problem of the complete classification of equations of this class remains open (see, for example, [15,16]).

(L1)

$u_{n+1,x}^j=u_{n,x}^j+e^{u_{n+1}^j-u_n^{j+1}}-e^{u_{n+1}^{j-1}-u_n^j},$

(L2)

$u_{n+1,x}^j=u_{n,x}^j+e^{u_{n+1}^{j-1}-u_{n+1}^j}-e^{u_{n+1}^{j-1}-u_n^j}-e^{u_n^j-u_n^{j+1}}+e^{u_{n+1}^j-u_n^{j+1}},$

(L3)

$u_{n+1,x}^j=u_{n,x}^j\frac{{\left(u_{n+1}^j\right)}^2}{u_{n+1}^{j-1}{u}_n^{j+1}},$

(L4)

$u_{n+1,x}^j=u_{n,x}^j\frac{u_{n+1}^j-u_n^{j+1}}{u_{n+1}^{j-1}-u_n^j},$

(L5)

$u_{n+1,x}^j=u_{n,x}^j\frac{u_{n+1}^j(u_{n+1}^j-u_n^{j+1})}{u_n^{j+1}(u_{n+1}^{j-1}-u_n^j)},$

(L6)

$u_{n+1,x}^j=u_{n,x}^j\frac{(u_{n+1}^{j-1}-u_{n+1}^j)(u_{n+1}^j-u_n^{j+1})}{(u_{n+1}^{j-1}-u_n^j)(u_n^j-u_n^{j+1})},$

(L7)

$u_{n+1,x}^j=u_{n,x}^j+e^{-u_n^{j+1}+u_n^j+u_{n+1}^j-u_{n+1}^{j-1}},$

(L8)

$u_{n+1,x}^j=u_{n,x}^j-e^{u_n^{j+1}}+e^{u_n^j}+e^{u_{n+1}^j}-e^{u_{n+1}^{j-1}},$

(L9)

$u_{n+1,x}^j=u_{n,x}^j+(u_{n+1}^j-u_n^j)(u_n^{j+1}-u_{n+1}^j-u_n^j+u_{n+1}^{j-1}).$

Equation (L1) coincides up to a simple change in the variables $\overline{u}(j,n)=e^{u_n^j}$ with Equation (N-7) in [7]. Equation (L8) in a slightly different form,

$$\frac{\partial }{\partial x_1}\left(v_{n,m+1}-v_{n+1,m}\right)=-\epsilon \left(e^{v_{n+1,m+1}}+e^{v_{n,m}}-e^{v_{n,m+1}}-e^{v_{n+1,m}}\right),$$

where $\epsilon =\frac{a-b}{ab}$, was also obtained in [7] (see (N-2)). Indeed, it is easy to check that it goes into (L8) after a point change in variables of the form

$$v_{n,m}=u_{-n+1}^{m+n-1},x_1=\frac{x}{\epsilon }.$$

Equation (L7) was found, to the best of our knowledge, in [8] as an equation for the Laplace invariants of the linear differential-difference equations of the hyperbolic type with variable coefficients. Equations (L2)–(L6) were obtained in [17] by an original method based on a preliminary study of the dispersionless limits of lattices for integrability, followed by a return transition to the class of lattices (1) as a result of dispersion deformations. Equation (L9) was found in [18].

A distinctive feature of nonlinear lattices with three independent variables is that they allow reductions in the form of systems of differential-difference equations with complete sets of characteristic integrals. Such systems of equations are called Darboux integrable. As a rule, it is possible to find in an explicit form general solutions to Darboux integrable systems at least for the case of small orders. This circumstance allows one to find localized particular solutions for the lattices in 3D via the Darboux integrable reductions. In our articles [15,19,20,21], an algorithm for classifying integrable equations of the form (1) is developed, where the presence of a hierarchy of the Darboux integrable reductions is taken as a classification criterion.

One can check that the equations from the list above are not related to each other by point changes in variables. However, the equations could be transformed from one to the other using more complex irreversible Miura-type transformations, which would be useful in finding explicit solutions to the equations (see, for instance, [22,23,24]). Note that the problem of the integrable classification of three-dimensional lattices is rather complicated and has not been solved so far, so the problem of finding new integrable equations by alternative methods, for example, using the Miura transformation, is topical. Therefore, the task of searching for such transformations is in demand. The study of this problem is one of the aims of the present work.

Let us briefly discuss the content of the article. In Section 2, we suggest an algorithm for searching differential or/and difference substitutions generated by a suitable representation of an equation that generalizes a local conservation law. In Section 3, we apply the algorithm to Equations (L1)–(L9). We establish relations between the equations. In Section 4, we give the Lax pair for lattice (L9) and calculate the continuum limit of two types for it. We also consider examples of the small-order Darboux integrable reductions admitted by the lattice. For the reduced systems, the complete sets of the characteristic integrals are constructed. An explicit particular solution to the lattice (L9) is presented, which is obtained via the reduced equation. A new example of an integrable equation with one continuous and two discrete variables (see (51)), which does not belong to the class of lattices (1), is found. The Lax pair for this model is presented.

2. Local Conservation Laws and Miura-Type Transformations

To find the Miura transformation that converts an equation of a given class (1) into some other equation of the same class, we use the following scheme. Let us first represent the equation as a local conservation law of the form below:

$$(D_n-1)A=(D_j-1)B$$

(2)

or the form

$$D_{x}A=(D_j-1)B,$$

(3)

where A and B are functions depending on the unknown function $u=u_n^j\left(x\right)$, as well as on its derivatives with respect to x and shifts with respect to discrete variables j and n. Here, $D_n$ and $D_j$ denote operators for shifting a function argument in n and j, for example, $D_{n}y\left(n\right)=y(n+1), D_{j}y\left(j\right)=y(j+1)$, and $D_x$ stands for the operator of the total differentiation with respect to x. If the representation (2) takes place, we define a new function $v=v_n^j\left(x\right)$ by introducing a potential, i.e., assuming

$$(D_j-1)v_n^j=A.$$

(4)

Then, substituting instead of the function A its representation (4) into relation (2), we arrive at the equality

$$(D_j-1)(D_n-1)v_n^j=(D_j-1)B,$$

(5)

from which, applying the operator inverse to $(D_j-1)$, we find another important relation connecting the variables $u_n^j$ and $v_n^j$:

$$(D_n-1)v_n^j=B+C.$$

(6)

Since we do concentrate only on autonomous transformations, we assume that C is a constant. In the examples considered below, it is usually possible to completely exclude the variable $u_n^j$ from the pair of relations (4) and (6). In such a case, we arrive at an equation for the new function $v_n^j$.

The second way to look for the desired substitution uses representation (3). Here, we introduce a new function ${\omega }_n^j$ due to the relation below:

$$D_x{\omega }_n^j=B.$$

(7)

Now, the equation yields a relation

$$D_{x}A=D_x\left(D_j-1\right){\omega }_n^j.$$

(8)

Integrating the latter, we find, in addition to (7), the following constraint that the variable ${\omega }_n^j$ should satisfy,

$$\left(D_j-1\right){\omega }_n^j=A+const.$$

(9)

To obtain the lattice equation for ${\omega }_n^j$, it is necessary to get rid of the variable $u_n^j$ by means of the relations (7) and (9).

As one can see from the examples below, this trick very often leads to success. In particular, it is in this way that we managed to find Miura-type transformations for each of the equations from the list under consideration.

Remark 1.

Note that some equations do not admit either of the representations (2) or (3) of the form of a conservation law. In this case, one can use a relation of a more general form (see Section 3.4).

3. Searching for the Transformations

In this section, we apply the algorithm discussed above to the lattices (L1)–(L9). It is interesting to note that the set of equations falls into two nonoverlapping subclasses. Most of the equations belong to the first subclass, consisting of different forms of the discrete version of 2D Toda lattice, namely (L1), (L3), (L4), (L7), (L8), and (L9). Only three equations fall into the second class: (L2), (L5), and (L6) (these equations were found in [17] and remain less studied). Equations from these two subclasses might be related to Miura-type transformations that are not of the form discussed in Section 2.

3.1. Relation between Equations (L1) and (L9)

Obviously, Equation (L1) can be rewritten as follows:

$$(D_n-1)u_{n,x}^j=(D_j-1)e^{u_{n+1}^{j-1}-u_n^j}.$$

(10)

According to the rule above, we introduce a new variable $v_n^j$ by setting

$$u_{n,x}^j=v_n^{j+1}-v_n^j=(D_j-1)v_n^j,$$

(11)

then Formulas (10) and (11) imply

$$(D_j-1)(D_n-1)v_n^j=(D_j-1)e^{u_{n+1}^{j-1}-u_n^j}$$

or the same

$$(D_n-1)v_n^j=e^{u_{n+1}^{j-1}-u_n^j}+C.$$

(12)

Let us differentiate (12) with respect to x and replace derivatives $u_{n+1,x}^{j-1}$, $u_{n,x}^j$ due to (11). As a result, we arrive at the equation

$$v_{n+1,x}^j-v_{n,x}^j=(v_{n+1}^j-v_{n+1}^{j-1}-v_n^{j+1}+v_n^j)(v_{n+1}^j-v_n^j-C),$$

(13)

that is reduced to (L9) by a point transformation $x=-\overline{x}$, $v_n^j=u_n^j+Cn$.

3.2. Relation between Equations (L2) and (L5)

Now, we derive a substitution generated by Equation (L5). To this end, we first simplify (L5) by the change in variables $u_n^j=e^{v_n^j}$:

$$v_{n+1,x}^j=v_{n,x}^j\frac{e^{v_{n+1}^j-v_n^{j+1}}-1}{e^{v_{n+1}^{j-1}-v_n^j}-1}.$$

Then, we represent the obtained equation in the form of the local conservation law:

$$(D_n-1)ln{v}_{n,x}^j=(D_j-1)ln\left(e^{v_{n+1}^{j-1}-v_n^j}-1\right).$$

(14)

Let us introduce a new dependent variable ${\omega }_n^j$ as

$$\left(D_j-1\right){\omega }_n^j=ln{v}_{n,x}^j$$

(15)

or

$$v_{n,x}^j=e^{{\omega }_n^{j+1}-{\omega }_n^j}.$$

Then, from (14) and (15) we obtain the equation

$$\left(D_j-1\right)\left(D_n-1\right){\omega }_n^j=\left(D_j-1\right)ln\left(e^{v_{n+1}^{j-1}-v_n^j}-1\right),$$

which implies the following:

$$\left(D_n-1\right){\omega }_n^j=ln\left(e^{v_{n+1}^{j-1}-v_n^j}-1\right)+C.$$

(16)

One can easily verify that the variable $v_n^j$ is completely eliminated from Formulas (15) and (16) and their consequences in a rather simple way. Indeed, by differentiating (16) we obtain

$${\omega }_{n+1,x}^j-{\omega }_{n,x}^j=\frac{e^{v_{n+1}^{j-1}-v_n^j}\left(v_{n+1,x}^{j-1}-v_{n,x}^j\right)}{e^{v_{n+1}^{j-1}-v_n^j}-1}.$$

(17)

The right-hand side of the formula is greatly simplified due to equalities (15) and (16), and, as a result, the equation reduces to the form

$${\omega }_{n+1,x}^j-{\omega }_{n,x}^j=\left(1+e^{{\omega }_n^j-{\omega }_{n+1}^j+C}\right)\left(e^{{\omega }_{n+1}^j-{\omega }_{n+1}^{j-1}}-e^{{\omega }_n^{j+1}-{\omega }_n^j}\right).$$

(18)

Up to the linear transformation ${\omega }_n^j=-u_n^j+Cn+\pi ij$, $i^2=-1$, Equation (18) coincides with Equation (L2).

3.3. Relation between Equations (L3) and (L1)

Let us turn to Equation (L3). Taking the logarithm of both sides of the equation, we obtain an identity

$$ln{u}_{n+1,x}^j-ln{u}_{n,x}^j=2ln{u}_{n+1}^j-ln{u}_{n+1}^{j-1}-ln{u}_n^{j+1},$$

(19)

which implies a local conservation law of the form

$$\left(D_n-1\right)ln\frac{u_{n,x}^j}{u_n^j}=\left(D_j-1\right)ln\frac{u_{n+1}^{j-1}}{u_n^j}.$$

(20)

Having introduced a new variable $h_n^j$ by setting

$$\left(D_j-1\right)h_n^j=ln\frac{u_{n,x}^j}{u_n^j},$$

(21)

we deduce from (20) the equality

$$\left(D_j-1\right)\left(D_n-1\right)h_n^j=\left(D_j-1\right)ln\frac{u_{n+1}^{j-1}}{u_n^j},$$

(22)

which gives rise to

$$\left(D_n-1\right)h_n^j=ln\frac{u_{n+1}^{j-1}}{u_n^j}+C,$$

(23)

where C is assumed to be constant. Now, we differentiate (23) with respect to x and find

$$h_{n+1,x}^j-h_{n,x}^j=\frac{u_{n+1,x}^{j-1}}{u_{n+1}^{j-1}}-\frac{u_{n,x}^j}{u_n^j},$$

which is easily simplified due to (21). As a result, we obtain the lattice

$$h_{n+1,x}^j-h_{n,x}^j=e^{h_{n+1}^j-h_{n+1}^{j-1}}-e^{h_n^{j+1}-h_n^j}.$$

The latter coincides with the discrete Toda lattice (L1) up to the linear change in the independent discrete variables $j=-\overline{j},n=\overline{j}+\overline{n}$.

3.4. Relation between Equations (L3) and (L7)

We note that for Equation (L3) one can derive one more substitution. To this end, we use Equation (19), rewritten as

$$\left(D_n-1\right)ln{u}_{n,x}^j=\left(2D_n-D_n{D}_j^{-1}-D_j\right)ln{u}_n^j.$$

(24)

Note that the relation (24) does not define any local conservation law.

Assuming that $ln{u}_n^j=\left(D_n-1\right)v_n^j$, we obtain

$$u_n^j=e^{v_{n+1}^j-v_n^j}.$$

(25)

Then, combining (24) and (25), we arrive at

$$u_{n,x}^j=e^{2v_{n+1}^j-v_{n+1}^{j-1}-v_n^{j+1}}$$

(26)

and by differentiating (25) we find a similar relation

$$u_{n,x}^j=\left(v_{n+1,x}^j-v_{n,x}^j\right)e^{v_{n+1}^j-v_n^j}.$$

(27)

Comparing the last two representations for $u_{n,x}^j$, one can obtain the equation

$$v_{n+1,x}^j-v_{n,x}^j=e^{-v_n^{j+1}+v_n^j+v_{n+1}^j-v_{n+1}^{j-1}},$$

(28)

which is nothing but Equation (L7).

3.5. Relation between Equations (L4) and (L1)

Now, we study Equation (L4) in the list. It can easily be represented as a local concervation law of the form

$$\left(D_n-1\right)ln{u}_{n,x}^j=\left(D_j-1\right)ln\left(u_{n+1}^{j-1}-u_n^j\right).$$

(29)

We introduce a new unknown $s_n^j$ by taking

$$\left(D_j-1\right)ln{s}_n^j=ln{u}_{n,x}^j.$$

(30)

Then, from (29) it follows that

$$\left(D_j-1\right)\left(D_n-1\right)ln{s}_n^j=\left(D_j-1\right)ln\left(u_{n+1}^{j-1}-u_n^j\right).$$

By applying to the latter equation the operator ${\left(D_j-1\right)}^{-1}$, we obtain the second part of the searched substitution:

$$\left(D_n-1\right)ln{s}_n^j=ln\left(u_{n+1}^{j-1}-u_n^j\right)+lnC,C=const.$$

(31)

Let us simplify (31) a little, after which, applying the operator $D_x$ to both its sides, we obtain a nonlinear lattice,

$$\frac{s_{n+1,x}^j}{s_n^j}-\frac{s_{n+1}^j{s}_{n,x}^j}{{\left(s_n^j\right)}^2}=C\left(\frac{s_{n+1}^j}{s_{n+1}^{j-1}}-\frac{s_n^{j+1}}{s_n^j}\right),$$

which is reduced to the discrete Toda lattice (L1) by the point transformation $s_n^j=e^{-u_n^j},$ $x\mapsto Cx$.

3.6. Relation between Equations (L6) and (L2)

Let us look for the Miura-type transformations generated by Equation (L6). By taking the logarithm of both sides of (L6), we obtain a relation,

$$\left(D_n-1\right)ln{u}_{n,x}^j=\left(D_n-D_j\right)ln\left(u_n^{j-1}-u_n^j\right)+\left(D_j-1\right)ln\left(u_{n+1}^{j-1}-u_n^j\right).$$

Afterwards, by using the decomposition $D_n-D_j=D_n-1-\left(D_j-1\right)$, one can rewrite the relation as a local conservation law of the form below:

$$\left(D_n-1\right)ln\frac{u_{n,x}^j}{u_n^{j-1}-u_n^j}=\left(D_j-1\right)ln\frac{u_{n+1}^{j-1}-u_n^j}{u_n^{j-1}-u_n^j}.$$

(32)

Then, we introduce a new variable $h_n^j$ by the rule

$$\left(D_j-1\right)ln{h}_n^j=ln\frac{u_{n,x}^j}{u_n^{j-1}-u_n^j}.$$

(33)

Applying the reasoning above to (32) and (33), one can derive:

$$\left(D_n-1\right)ln{h}_n^j=ln\frac{u_{n+1}^{j-1}-u_n^j}{u_n^{j-1}-u_n^j}+lnC.$$

(34)

Let us rewrite Miura transformations (33) and (34) in a more convenient form

$$\frac{u_{n,x}^j}{u_n^{j-1}-u_n^j}=\frac{h_n^{j+1}}{h_n^j},C\frac{u_{n+1}^{j-1}-u_n^j}{u_n^{j-1}-u_n^j}=\frac{h_{n+1}^j}{h_n^j},$$

(35)

which implies that

$$\frac{1}{C}\frac{u_{n,x}^j}{u_{n+1}^{j-1}-u_n^j}=\frac{h_n^{j+1}}{h_{n+1}^j}.$$

(36)

Differentiating (34), we find a relation

$$\begin{array}{cc}\hfill \frac{h_{n+1,x}^j}{h_{n+1}^j}-\frac{h_{n,x}^j}{h_n^j}=& \frac{u_{n+1,x}^{j-1}-u_{n,x}^j}{u_{n+1}^{j-1}-u_n^j}-\frac{u_{n,x}^{j-1}-u_{n,x}^j}{u_n^{j-1}-u_n^j}=\hfill \\ \hfill =& \frac{u_{n+1,x}^{j-1}}{u_{n+1}^{j-1}-u_n^j}-C\frac{h_n^{j+1}}{h_{n+1}^j}-\frac{u_{n,x}^{j-1}}{u_n^{j-1}-u_n^j}+\frac{h_n^{j+1}}{h_n^j}.\hfill \end{array}$$

(37)

We simplify some of the terms in (37) due to lattice (L6) and obtain the following:

$$\begin{array}{cc}\hfill & \frac{u_{n+1,x}^{j-1}}{u_{n+1}^{j-1}-u_n^j}-\frac{u_{n,x}^{j-1}}{u_n^{j-1}-u_n^j}=\frac{u_{n,x}^{j-1}}{u_n^{j-1}-u_n^j}\left(\frac{u_{n+1}^{j-2}-u_{n+1}^{j-1}}{u_{n+1}^{j-2}-u_n^{j-1}}-1\right)=\hfill \\ \hfill & =C\frac{h_n^j}{h_{n+1}^{j-1}}\frac{u_n^{j-1}-u_{n+1}^{j-1}}{u_n^{j-1}-u_n^j}=C\frac{h_n^j}{h_{n+1}^{j-1}}\left(1-\frac{u_{n+1}^{j-1}-u_n^j}{u_n^{j-1}-u_n^j}\right)=C\frac{h_n^j}{h_{n+1}^{j-1}}\left(1-\frac{1}{C}\frac{h_{n+1}^j}{h_n^j}\right).\hfill \end{array}$$

Finally, we obtain a lattice,

$$\frac{h_{n+1,x}^j}{h_{n+1}^j}-\frac{h_{n,x}^j}{h_n^j}=\frac{h_n^{j+1}}{h_n^j}-C\frac{h_n^{j+1}}{h_{n+1}^j}+C\frac{h_n^j}{h_{n+1}^{j-1}}-\frac{h_{n+1}^j}{h_{n+1}^{j-1}},$$

which is related to lattice (L2) by a point transformation $h_n^j=C^n{e}^{-p_n^j}$:

$$p_{n+1,x}^j-p_{n,x}^j=e^{p_{n+1}^{j-1}-p_{n+1}^j}-e^{p_{n+1}^{j-1}-p_n^j}-e^{p_n^j-p_n^{j+1}}+e^{p_{n+1}^j-p_n^{j+1}}.$$

3.7. Relation between Equations (L8) and (L1)

Let us look for a substitution that would connect lattices (L8) and (L1). At first, we rewrite Equation (L8) in the form of a local conservation law:

$$\left(D_n-1\right)u_{n,x}^j=\left(D_j-1\right)\left(e^{u_{n+1}^{j-1}}-e^{u_n^j}\right).$$

(38)

Supposing that

$$u_{n,x}^j=\left(D_j-1\right)p_{n,x}^j,$$

(39)

we introduce a new dependent variable $p_n^j$. Then, obviously, it follows from (38) and (39) that

$$\left(D_j-1\right)\left(D_n-1\right)p_{n,x}^j=\left(D_j-1\right)\left(e^{u_{n+1}^{j-1}}-e^{u_n^j}\right).$$

(40)

From the latter, one can derive the second part of the Miura-type transformation:

$$\left(D_n-1\right)p_{n,x}^j=e^{u_{n+1}^{j-1}}-e^{u_n^j}+C_1.$$

(41)

The expression $u_n^j=\left(D_j-1\right)p_n^j+C_2$ is an evident consequence of (39); therefore, we have

$$e^{u_n^j}=e^{C_2}{e}^{p_n^{j+1}-p_n^j}.$$

(42)

Let us eliminate the variables $u_n^j,u_{n+1}^{j-1}$ from (41) due to (42) and obtain a lattice of the form below:

$$p_{n+1,x}^j-p_{n,x}^j=e^{C_2}\left(e^{p_{n+1}^j-p_{n+1}^{j-1}}-e^{p_n^{j+1}-p_n^j}\right)+C_1.$$

Changing the variables by setting $p_n^j=q_n^j+C_{1}xn$, we arrive at the lattice

$$q_{n+1,x}^j-q_{n,x}^j=e^{C_2}\left(e^{q_{n+1}^j-q_{n+1}^{j-1}}-e^{q_n^{j+1}-q_n^j}\right),$$

which converts to the discrete Toda lattice (L1) under replacement $j=-\overline{j}$, $n=\overline{j}+\overline{n}$, $x=e^{-C_2}\overline{x}$.

4. Investigation of Lattice (L9)

In this section, we focus on a detailed study of lattice (L9): we calculate the continuum limit, study Darboux integrable reductions, construct explicit solutions, and show that it defines a Bäcklund transformation for an equation of the Toda lattice type. For reduced systems of small dimensions, characteristic integrals are found in both directions n and x. We also briefly discuss the method of constructing particular solutions of lattice (L9) by means of its reduction.

Let us begin with the Lax pair. It is checked straightforwardly that the system of linear equations [18]

$$\begin{array}{cc}\hfill & {\psi }_{n+1}^j=\left(v_n^j-v_{n+1}^j\right){\psi }_n^{j+1}+{\psi }_n^j,\hfill \\ \hfill & {\psi }_{n,x}^j=\left(v_n^j-v_n^{j-1}\right){\psi }_n^j-{\psi }_n^{j-1}\hfill \end{array}$$

(43)

is compatible if and only if function $v=v_n^j\left(x\right)$ is a solution to lattice (L9).

4.1. Computation of the Continuum Limit for Lattice (L9)

In this section, we will calculate the continuum limit of lattice (L9) under the unbounded refinement of the difference grid in both directions n and j. At first, we perform a scaling transformation $y={\delta }^{2}x$ in lattice (51) and represent it as follows:

$$\begin{array}{cc}\hfill & {\delta }^2\frac{d}{dy}\left(v_{n+1}^j-v_n^j\right)=\hfill \\ \hfill & =\left(v_{n+1}^j-v_n^j\right)\left[\left(v_n^{j+1}-2v_n^j+v_n^{j-1}\right)+\left(v_{n+1}^{j-1}-v_n^{j-1}\right)-\left(v_{n+1}^j-v_n^j\right)\right].\hfill \end{array}$$

(44)

Let us assume that the unknown function $v_n^j\left(x\right)$ is of the form $v_n^j\left(x\right)=u(t,z,y)$, where $t={\delta }^{2}n$, $z=\delta j$, and $\delta$ is a small positive parameter. Now, we estimate the difference derivatives with respect to the variables n and j appearing in (44) by virtue of the Taylor formula:

$$v_{n+1}^j-v_n^j=u(t+{\delta }^2,z,y)-u(t,z,y)={\delta }^2{u}_t(t,z,y)+O\left({\delta }^4\right),\delta \mapsto 0,$$

(45)

$$\begin{array}{cc}\hfill v_{n+1}^{j-1}-v_n^{j-1}& =u(t+{\delta }^2,z-\delta ,y)-u(t,z-\delta ,y)=\hfill \\ \hfill & =u(t,z,y)+{\delta }^2{u}_t(t,z,y)-\delta u_z(t,z,y)+\frac{1}{2}{\delta }^2{u}_{zz}(t,z,y)-\hfill \\ \hfill & -u(t,z,y)+\delta u_z(t,z,y)-\frac{1}{2}{\delta }^2{u}_{zz}(t,z,y)+\dots +O\left({\delta }^3\right)=\hfill \\ \hfill & ={\delta }^2{u}_t(t,z,y)+O\left({\delta }^3\right),\delta \mapsto 0,\hfill \end{array}$$

$$\begin{array}{cc}\hfill v_n^{j+1}-2v_n^j+v_n^{j-1}& =u(t,z+\delta ,y)-2u(t,z,y)+u(t,z-\delta ,y)=\hfill \\ \hfill & =u(t,z,y)+\delta u_z(t,z,y)+\frac{1}{2}{\delta }^2{u}_{zz}(t,z,y)-2u(t,z,y)+\hfill \\ \hfill & +u(t,z,y)-\delta u_z(t,z,y)+\frac{1}{2}{\delta }^2{u}_{zz}(z,t,y)+\dots =\hfill \\ \hfill & ={\delta }^2{u}_{zz}(t,z,y)+O\left({\delta }^3\right),\delta \mapsto 0.\hfill \end{array}$$

Now, we substitute the found asymptotic representations of finite differences into Equation (44) and pass to the limit at $\delta \mapsto 0$. As a result, we obtain the following dispersionless partial differential equation:

$$u_{ty}=u_t{u}_{zz},$$

(46)

which is integrable as well [25].

4.2. Intermediate Continuum Limit and Bäcklund Transformation—A New Integrable Lattice

Now, we study the continuum limit of lattice (L9), assuming that unknown function $u_n^j\left(x\right)$ depends on a small parameter $\epsilon >0$ as follows $u_n^j\left(x\right)={\omega }^j(x,t)$ with $t=\epsilon n$. Under the condition $\epsilon \mapsto 0$, lattice (L9) tends to an equation of the Toda lattice type [23]

$${\omega }_{xt}^j+{\omega }_t^j\left({\omega }^{j+1}-2{\omega }^j+{\omega }^{j-1}\right)=0$$

(47)

with a Lax pair of the form

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\phi }_t^j=-{\omega }_t^j{\phi }^{j+1}+{\phi }^j,\hfill \\ \hfill & {\phi }_x^j=\left({\omega }^j-{\omega }^{j-1}\right){\phi }^j-{\phi }^{j-1}\hfill \end{array}\end{array}$$

(48)

which is easily obtained from (43) under the relation ${\psi }_n^j\left(x\right)={\phi }^j(x,t),$ when $\epsilon \mapsto 0$. A remarkable fact is that lattices (L9) and (47) have a close connection. Indeed, by comparing the Lax pairs one can conclude that lattice (L9) realizes a sequence of the Bäcklund transformations for lattice (47) (to the best of our knowledge, this observation is new). More precisely, any solution ${\omega }^j$ of (47) is mapped by the Bäcklund transformation

$$\begin{array}{cc}\hfill & {\overline{\omega }}_x^j-{\omega }_x^j=\left({\overline{\omega }}^j-{\omega }^j\right)\left(-{\omega }^{j+1}+{\omega }^j+{\overline{\omega }}^j-{\overline{\omega }}^{j-1}\right),\hfill \\ \hfill & \left({\omega }_t^{j+1}+{\overline{\omega }}_t^{j-1}\right)\left({\overline{\omega }}^j-{\omega }^j\right)+{\overline{\omega }}_t^j\left({\omega }^{j+1}-{\overline{\omega }}^{j+1}\right)+{\omega }_t^j\left({\omega }^{j-1}-{\overline{\omega }}^{j-1}\right)=0\hfill \end{array}$$

(49)

to another solution ${\overline{\omega }}^j$ of the same lattice.

We note that eigenfunctions ${\phi }^j$ and ${\overline{\phi }}^j$ of the system (48) and of the system

$${\overline{\phi }}_t^j=-{\overline{\omega }}_t^j{\overline{\phi }}^{j+1}+{\overline{\phi }}^j,{\overline{\phi }}_x^j=\left({\overline{\omega }}^j-{\overline{\omega }}^{j-1}\right){\overline{\phi }}^j-{\overline{\phi }}^{j-1}$$

are connected with one another as follows:

$${\overline{\phi }}^j=\left({\omega }^j-{\overline{\omega }}^j\right){\phi }^{j+1}+{\phi }^j.$$

(50)

Iterations of the Bäcklund transformation (49) determine two nonlinear equations of the form

$$\begin{array}{cc}\hfill & \left(i\right)w_{n+1,x}^j=w_{n,x}^j+(w_{n+1}^j-w_n^j)(-w_n^{j+1}+w_{n+1}^j+w_n^j-w_{n+1}^{j-1}),\hfill \\ \hfill & \left(ii\right)\left({\omega }_{n,t}^{j+1}+{\omega }_{n+1,t}^{j-1}\right)\left({\omega }_{n+1}^j-{\omega }_n^j\right)+\hfill \\ \hfill & +{\omega }_{n+1,t}^j\left({\omega }_n^{j+1}-{\omega }_{n+1}^{j+1}\right)+{\omega }_{n,t}^j\left({\omega }_n^{j-1}-{\omega }_{n+1}^{j-1}\right)=0.\hfill \end{array}$$

(51)

The first lattice coincides with (L9) up to substitution $x\to -x$; the second one does not belong to the class (1). It is also integrable, since it admits the Lax pair

$$\begin{array}{cc}\hfill & {\phi }_{n+1}^j=\left({\omega }_n^j-{\omega }_{n+1}^j\right){\phi }^{j+1}+{\phi }_n^j,\hfill \\ \hfill & {\phi }_{n,t}^j=-{\omega }_{n,t}^j{\phi }_n^{j+1}+{\phi }_n^j.\hfill \end{array}$$

Note that integrable lattices of this type have not been previously considered in the literature.

4.3. Darboux Integrable Reductions of Lattice (L9)

Lattice (L9) admits the following cutting-off boundary condition

$${v_n^j\left(x\right)|}_{j=j_0}=C,C=const$$

(52)

preserving the integrability property. Imposing this kind of the condition on two points $j=-1$ and $j=N$, we obtain a finite field reduction of lattice (L9):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & v_n^{-1}=C_0,\hfill \\ \hfill & v_{n+1,x}^j-v_{n,x}^j=\left(v_{n+1}^j-v_n^j\right)\left(-v_n^{j+1}+v_n^j+v_{n+1}^j-v_{n+1}^{j-1}\right),0\le j\le N-1,\hfill \\ \hfill & v_n^N=C_1,\hfill \end{array}\end{array}$$

(53)

admitting the Lax representation of the form

$$\begin{array}{cc}\hfill & \left\{\begin{array}{c}{\psi }_{n+1}^j=\left(v_n^j-v_{n+1}^j\right){\psi }_n^{j+1}+{\psi }_n^j,0\le j\le N-1,\hfill \\ {\psi }_{n+1}^N={\psi }_n^N,\hfill \end{array}\right.\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill & \left\{\begin{array}{c}{\psi }_{n,x}^0=\left(v_n^0-C_0\right){\psi }_n^0,\hfill \\ {\psi }_{n,x}^j=\left(v_n^j-v_n^{j-1}\right){\psi }_n^j-{\psi }_n^{j-1},1\le j\le N\hfill \end{array}\right.\hfill \end{array}$$

(55)

deduced from (43) under the constraints ${\psi }_n^{-1}={\psi }_n^{N+1}=0$, $v_n^{-1}=C_0$, $v_n^N=C_1$. Let us concentrate on the finite field system (53). It is worth noting that (53) admits complete sets of integrals in both directions n and x. Recall that a function I, depending on the dynamical variables, is an n-integral of the differential-difference system if it satisfies the relation $D_{n}I=I$ by virtue of the system. Similarly, a function J depending on dynamical variables is an x-integral if it solves the equation $D_{x}J=0$ by means of the system. Integrals of the form $I=I\left(x\right)$ and $J=J\left(n\right)$ depending on x and n only are called trivial. We are only interested in nontrivial integrals. Let us give examples of integrals.

Example 1.

By setting $N=1$ in (53), we obtain a scalar equation for $v:=v^0$

$$v_{n+1,x}-v_{n,x}=\left(v_{n+1}-v_n\right)\left(v_{n+1}+v_n-C_0-C_1\right),$$

(56)

which admits n-integral

$$I=v_x-{(v-C_0)}^2-(v-C_1)(C_0-C_1)$$

(57)

and x-integral

$$J=\frac{\left(v_n-v_{n+2}\right)\left(v_{n+1}-v_{n+3}\right)}{\left(v_n-v_{n+3}\right)\left(v_{n+1}-v_{n+2}\right)}.$$

(58)

Example 2.

For the choice $N=2$, we arrive at the system for $u_n=v_n^0$, $v_n=v_n^1$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & u_{n+1,x}-u_{n,x}=\left(u_{n+1}-u_n\right)\left(-v_n+u_n+u_{n+1}-C_0\right),\hfill \\ \hfill & v_{n+1,x}-v_{n,x}=\left(v_{n+1}-v_n\right)\left(-C_1+v_n+v_{n+1}-u_{n+1}\right).\hfill \end{array}\end{array}$$

(59)

The obtained system admits a complete set of n-integrals, i.e., a set of two independent n-integrals

$$\begin{array}{cc}\hfill & I_1=u_{n,x}+v_{n,x}-{\left(u_n-C_0\right)}^2-\left(u_n-C_1\right)\left(C_0-v_n\right)-{\left(C_1-v_n\right)}^2,\hfill \\ \hfill & I_2=v_{n,xx}-2\left(v_n-C_1\right)v_{n,x}-\left(C_0-v_n\right)u_{n,x}-\left(v_n-C_0\right)\left(u_n-C_1\right)\left(u_n-v_n-C_0+C_1\right)\hfill \end{array}$$

and a complete set of x-integrals

$$\begin{array}{cc}\hfill & J_1=\frac{\left({\tau }_n^0{\tau }_n^1+{\tau }_{n+1}^0{\tau }_{n+1}^1+{\tau }_n^0{\tau }_{n+1}^1\right)({\tau }_{n+1}^0{\tau }_{n+1}^1+{\tau }_{n+2}^0{\tau }_{n+2}^1+{\tau }_{n+1}^0{\tau }_{n+2}^1)}{{\tau }_n^0{\tau }_{n+2}^1\left({\tau }_{n+1}^0{\tau }_n^1+{\tau }_{n+2}^0{\tau }_{n+1}^1+{\tau }_{n+2}^0{\tau }_n^1\right)},\hfill \\ \hfill & J_2=\frac{{\tau }_{n+3}^0{\tau }_n^1\left({\tau }_{n+1}^0{\tau }_{n+1}^1+{\tau }_{n+2}^0{\tau }_{n+2}^1+{\tau }_{n+1}^0{\tau }_{n+2}^1\right)}{{\tau }_{n+2}^0{\tau }_{n+1}^1\left({\tau }_{n+1}^0{\tau }_n^1+{\tau }_{n+2}^0{\tau }_{n+1}^1+{\tau }_{n+2}^0{\tau }_n^1+{\tau }_{n+3}^0{\tau }_{n+2}^1+{\tau }_{n+3}^0{\tau }_{n+1}^1+{\tau }_{n+3}^0{\tau }_n^1\right)},\hfill \end{array}$$

where ${\tau }_n^0=u_{n+1}-u_n$, ${\tau }_n^1=v_{n+1}-v_n$ (about the complete set of integrals, see, for instance, [26,27]).

It can be proved that system (53) admits complete sets of x-integrals and n-integrals for any choice of N. For $N\ge 3$, x-integrals have extremely large expressions. They are found by the method of characteristic Lie–Rinehart algebras (see [21]). As can be seen from the above examples, the n-integrals of the reduced system are much simpler. Note that the complete set of n-integrals to the system (53) for an arbitrary natural $N\ge 1$ can be efficiently derived using the algorithm proposed in Lemma 2 of [21]. An alternative approach to obtain integrals is proposed in [28].

4.4. Explicit Solutions to Lattice (L9)

The existence of complete sets of integrals in both characteristic directions x and n makes it possible to completely separate the variables. Indeed, according to the definition, each solution $v=(v^0,v^1,\cdots ,v^{N-1})$ of system (53) simultaneously satisfies two systems of equations, namely the system of ordinary differential equations

$$I_1={\phi }_1\left(x\right),I_2={\phi }_2\left(x\right),\dots ,I_N={\phi }_N\left(x\right)$$

(60)

and the system of ordinary discrete equations

$$J_1={\psi }_1\left(n\right),J_2={\psi }_2\left(n\right),\dots ,J_N={\psi }_N\left(n\right),$$

(61)

where ${\phi }_1\left(x\right),{\phi }_2\left(x\right),\dots ,{\phi }_N\left(x\right)$ are arbitrary functions of x and similarly ${\psi }_1\left(n\right),{\psi }_2\left(n\right),\dots ,$${\psi }_N\left(n\right)$ are arbitrary functions of the variable n.

In this case, the common solution ${\left\{v_n^j\left(x\right)\right\}}_{j=0}^{N-1}$ of the systems provides a solution to the system (53). We remark that the solution of the reduced system (53) can obviously be prolonged to a particular solution of the original lattice (L9). However, for arbitrary N, system (60) is rather difficult to solve.

Here, we discuss a comparatively simple case, $N=1$. Let us derive a formula describing the general solution to lattice (56). Due to the formula $D_{n}I=I$, one can conclude that $I=\phi \left(x\right)$, i.e., function $v_n\left(x\right)$ is a solution to the equation

$$v_{n,x}-{\left(v_n-C_0\right)}^2-\epsilon \left(v_n-C_1\right)=\phi \left(x\right),\epsilon =C_0-C_1.$$

(62)

Since $\phi \left(x\right)$ is arbitrary, we can assume that $\phi \left(x\right)$ is represented as

$$\phi \left(x\right)=z_x-{(z-C_0)}^2-\epsilon (z-C_1)$$

for function $z=z\left(x\right)$, which is now considered a new functional parameter instead of $\phi \left(x\right)$. As a result, we obtain

$$v_{n,x}-{\left(v_n-C_0\right)}^2-\epsilon \left(v_n-C_1\right)=z_x-{(z-C_0)}^2-\epsilon (z-C_1).$$

(63)

Now, we have Equation (63), where $z=z\left(x\right)$ is an arbitrary given function of x and $v_n=v_n\left(x\right)$ is an unknown function. Let us make a change in the variables in Equation (63) by setting $v_n\left(x\right)=z\left(x\right)+s_n\left(x\right)$. Then, we obtain

$$s_{n,x}-s_n^2-s_n\left(\epsilon +2\left(z-C_0\right)\right)=0,$$

(64)

which has a form of the Bernoulli equation. Let us replace $s_n\left(x\right)=\frac{1}{g_n\left(x\right)}$ and arrive at a linear inhomogeneous equation:

$$g_{n,x}=-g_n\left(\epsilon +2\left(z-C_0\right)\right)-1.$$

(65)

In order to solve the corresponding homogeneous equation

$$g_{n,x}=-g_n\left(\epsilon +2\left(z-C_0\right)\right)$$

we again change the functional parameter. We pass from the functional parameter $z=z\left(x\right)$ to a new parameter $p\left(x\right)$ in such a way that

$$p_x=\epsilon +2(z-C_0).$$

Then, the homogeneous equation

$$g_{n,x}=-g_n{p}_x$$

is easily integrated into $g_n\left(x\right)=C{e}^{-p\left(x\right)}$. Now, supposing $C=C\left(x\right)$, we return to Equation (65) and find $C_x=-e^{p\left(x\right)}$. We change the parameter in the equation again by setting $e^{p\left(x\right)}={\omega }_x$, where $\omega =\omega \left(x\right)$ is a new parameter. As a result, we obtain $C\left(x\right)=-\omega \left(x\right)+K\left(n\right)$, where $K\left(n\right)$ is a constant of integration that might depend on n. Therefore, we have

$$g_n=\frac{K\left(n\right)-\omega }{{\omega }_x}\mathrm{so} \mathrm{that}{s}_n\left(x\right)=\frac{{\omega }_x}{K\left(n\right)-\omega }.$$

To write down the final form of $v_n\left(x\right)$, we need an expression of $z\left(x\right)$ in terms of $\omega \left(x\right)$. To this end, we exclude parameter $p\left(x\right)$ from the relations

$$z\left(x\right)=C_0+\frac{1}{2}(p_x-\epsilon ),e^{p\left(x\right)}={\omega }_x$$

and find

$$z\left(x\right)=\frac{1}{2}\frac{{\omega }_{xx}}{{\omega }_x}+\frac{C_0+C_1}{2}.$$

Now, we substitute the found expressions into $v_n\left(x\right)=s_n\left(x\right)+z\left(x\right)$ and obtain a general solution to lattice (56):

$$v_n\left(x\right)=\frac{1}{2}\frac{{\omega }_{xx}}{{\omega }_x}+\frac{{\omega }_x}{K\left(n\right)-\omega }+\frac{C_0+C_1}{2},$$

(66)

where $\omega =\omega \left(x\right)$ is an arbitrary function of x and $K\left(n\right)$ is an arbitrary function of n. By direct calculation, one can verify that function (66) is a solution to lattice (56) for any functions $\omega \left(x\right)$ and $K\left(n\right)$. In a more general situation, to clarify the form of the solution, it is also necessary to use integrals in another direction.

Due to the fact that the termination conditions $v_n^{-1}=C_0$, $v_n^1=C_1$ are fully consistent with the dynamics, by virtue of the original three-dimensional lattice (L9), the solution of the reduced system (56) extends in an obvious way to the solution of lattice (L9):

$$\begin{array}{c}\hfill v_n^j\left(x\right)=\left\{\begin{array}{cc}{C}_0\hfill & \mathrm{for}j\le -1,\hfill \\ {\displaystyle \frac{{\omega }_{xx}}{2{\omega }_x}+\frac{{\omega }_x}{K\left(n\right)-\omega }+\frac{C_0-C_1}{2}}\hfill & \mathrm{for}j=0,\hfill \\ C_1\hfill & \mathrm{for}j\ge 1.\hfill \end{array}\right.\end{array}$$

A problem of constructing explicit solutions via characteristic integrals is discussed in [29,30,31]. The papers convince that this approach is effective but needs further development, since the existing algorithms are technically complex.

5. Conclusions

In this article, integrable lattices with three independent variables, x, n, and j, are considered. A list of currently known examples is given. Differential and difference substitutions linking equations of the list with each other are presented. One of the lattices is studied in more detail. For that one, the continuum limits are computed. The degenerate cutoff condition is pointed out, which reduces the lattice to a Darboux integrable finite system of hyperbolic-type differential-difference equations when it is imposed on two different points, $(x,n,j=0)$ and $(x,n,j=N)$. For the reductions of small orders, characteristic integrals are found. A method for constructing explicit particular solutions to the lattice via reductions is proposed.

A new integrable lattice is found that is not of the form (1). For this model, the Lax pair is presented.