A group theory approach towards some rational difference equations

TL;DR

This paper applies Lie symmetry analysis to rational difference equations, deriving non-trivial symmetries and obtaining exact solutions, advancing the understanding of their structural properties.

Contribution

It introduces a group theory approach to analyze rational difference equations, providing a systematic method to find symmetries and solutions.

Findings

Derived non-trivial Lie point symmetries for rational difference equations

Obtained exact solutions using symmetry methods

Enhanced understanding of the structural properties of these equations

Abstract

A full Lie point symmetry analysis of rational difference equations is performed. Non-trivial symmetries are derived and exact solutions using these symmetries are obtained.

Equations

$$x_{n+1}=\frac{x_{n-5}}{\pm 1\pm x_{n-1}{x}_{n-3}{x}_{n-5}},$$

$$x_{n+1}=\frac{x_{n-5}}{a_n+b_n{x}_{n-1}{x}_{n-3}{x}_{n-5}},$$

$$u_{n+6}=\frac{u_n}{A_n+B_n{u}_n{u}_{n+2}{u}_{n+4}}$$

$$x_{n+1}=\frac{x_{n-4}{x}_{n-3}{x}_{n-2}}{x_{n-1}{x}_n(\lambda +\mu x_{n-4}{x}_{n-3}{x}_{n-2})},$$

$$u_{n+5}=\frac{u_n{u}_{n+1}{u}_{n+2}}{u_{n+3}{u}_{n+4}(\lambda +\mu u_n{u}_{n+1}{u}_{n+2})},$$

$${\Gamma }_{\epsilon }:x\mapsto \hat{x}(x;\epsilon ),$$

$$\widehat{x_i}(x;\epsilon )=x_i+\epsilon {\xi }_i(x)+O({\epsilon }^2),i=1,...,p.$$

$$u_{n+p}=\Omega (n,u_n,\dots ,u{}_{n+p-1}),$$

$$\hat{n}=n;{\hat{u}}_n=u_n+\epsilon Q(n,u_n)+O({\epsilon }^2)$$

$$\begin{array}{rl}X=& Q(n,u_n)\frac{\partial }{\partial u_n}+SQ(n,u_n)\frac{\partial }{\partial u_{n+1}}+\cdots +S^{(p-1)}Q(n,u_n)\frac{\partial }{\partial u_{n+p-1}},\end{array}$$

$${\hat{u}}_{n+p}=\Omega (n,{\hat{u}}_n,{\hat{u}}_{n+1},...,{\hat{u}}_{n+p-1}),$$

$$S^{(p)}Q-X\Omega =0$$

$${\Theta }^n({\theta }_j,s,l)=\prod _{j=s}^l{\theta }_{2j+n},n=0,1,$$

$$\prod _{j=s}^l{\theta }_j=1\text{ when }s>l\text{ and }\sum _{j=n}^{l_0}{\theta }_j=0\text{ when }n>l_0.$$

$$u_{n+6}=\frac{u_n}{A_n+B_n{u}_n{u}_{n+2}{u}_{n+4}}.$$

$$S^{6}Q+\frac{B_n{u_n}^2{u}_{n+2}{S}^{4}Q+B_n{u_n}^2{u}_{n+4}{S}^{2}Q-A_{n}Q}{(B_n{u}_n{u}_{n+2}{u}_{n+4}+A_n{)}^2}=0.$$

$$\begin{array}{rl}& S^{4}Q+u_{n+4}(S^{2}Q{)}^{\prime }-u_{n+4}{Q}^{\prime }+\frac{2u_{n+4}}{u_n}Q=0,\end{array}$$

$$\begin{array}{rl}& Q^{\prime \prime }\left(n,u_n\right)-\frac{2}{u_n}{Q}^{\prime }\left(n,u_n\right)+\frac{2}{u_n^2}Q\left(n,u_n\right)=0.\end{array}$$

$$\begin{array}{r}Q\left(n,u_n\right)={\alpha }_n{u_n}^2+{\beta }_n{u}_n\end{array}$$

$$u_n{u}_{n+2}{u_{n+4}}^2$$

$$u_n{u_{n+2}}^2{u}_{n+4}$$

$$u_n{u}_{n+2}{u}_{n+4}$$

$$u_n$$

$$1$$

$${\alpha }_n=0,{\beta }_{n+4}+{\beta }_{n+2}+{\beta }_n=0.$$

$${\alpha }_n=0\text{ and }{\beta }_n=(-1{)}^n{\beta }^n{c}_1+(-1{)}^n{\bar{\beta }}^n{c}_2+{\bar{\beta }}^n{c}_3+{\beta }^n{c}_4$$

$$Q_1=(-1{)}^n{\beta }^n{u}_n,Q_2=(-1{)}^n{\bar{\beta }}^n{u}_n{Q}_3={\bar{\beta }}^n{u}_n,Q_4={\beta }^n{u}_n.$$

$$\begin{array}{rl}{X}_1=& (-1{)}^n{\beta }^n{u}_n\partial u_n-(-1{)}^n{\beta }^{n+1}{u}_{n+1}{\partial }_{u_{n+1}}+(-1{)}^n{\beta }^{n+2}{u}_{n+2}{\partial }_{u_{n+2}}-\\ & (-1{)}^n{\beta }^{n+3}{u}_{n+3}{\partial }_{u_{n+3}}+(-1{)}^n{\beta }^{n+4}{u}_{n+4}{\partial }_{u_{n+4}}-(-1{)}^n{\beta }^{n+5}{u}_{n+5}{\partial }_{u_{n+5}},\end{array}$$

$$\begin{array}{rl}{X}_2=& (-1{)}^n{\bar{\beta }}^n{u}_n\partial u_n-(-1{)}^n{\bar{\beta }}^{n+1}{u}_{n+1}{\partial }_{u_{n+1}}+(-1{)}^n{\bar{\beta }}^{n+2}{u}_{n+2}{\partial }_{u_{n+2}}-\\ & (-1{)}^n{\bar{\beta }}^{n+3}{u}_{n+3}{\partial }_{u_{n+3}}+(-1{)}^n{\bar{\beta }}^{n+4}{u}_{n+4}{\partial }_{u_{n+4}}-(-1{)}^n{\bar{\beta }}^{n+5}{u}_{n+5}{\partial }_{u_{n+5}},\end{array}$$

$$\begin{array}{rl}{X}_3=& {\bar{\beta }}^n{u}_n\partial u_n+{\bar{\beta }}^{n+1}{u}_{n+1}{\partial }_{u_{n+1}}+{\bar{\beta }}^{n+2}{u}_{n+2}{\partial }_{u_{n+2}}+{\bar{\beta }}^{n+3}{u}_{n+3}{\partial }_{u_{n+3}}+\\ & {\bar{\beta }}^{n+4}{u}_{n+4}{\partial }_{u_{n+4}}+{\bar{\beta }}^{n+5}{u}_{n+5}{\partial }_{u_{n+5}},\end{array}$$

$$\begin{array}{rl}{X}_4=& {\beta }^n{u}_n\partial u_n+{\beta }^{n+1}{u}_{n+1}{\partial }_{u_{n+1}}+{\beta }^{n+2}{u}_{n+2}{\partial }_{u_{n+2}}+{\beta }^{n+3}{u}_{n+3}{\partial }_{u_{n+3}}+\\ & {\beta }^{n+4}{u}_{n+4}{\partial }_{u_{n+4}}+{\beta }^{n+5}{u}_{n+5}{\partial }_{u_{n+5}}.\end{array}$$

$$s_n=\int \frac{d{u}_n}{{\beta }^n{u}_n}=\frac{1}{{\beta }^n}\ln |u_n|.$$

$$r_n={\beta }^{n+4}{s}_{n+4}+{\beta }^{n+2}{s}_{n+2}+{\beta }^n{s}_n.$$

$$X_i{r}_n={\beta }^{n+4}+{\beta }^{n+2}+{\beta }^n=0,i=1,\dots ,4.$$

Full text

Keywords: Difference equation; symmetry; canonical coordinates; group invariant solutions

MSC: 39A10; 39A13; 39A99

1 Introduction

Over a century ago, symmetries became a centre of interest of several authors after the work of Sophus Lie [6] on differential equations. He studied the continuous group of transformations that leaves the differential equations invariant. This concept of symmetries is strongly related to the existence of conservation laws and the relationship between them has attracted great interest among researchers following the work of Noether [7]. The extension of this idea to difference equations is now well-documented (see [4] and references herein). In [4], Hydon developed a symmetry based algorithm enabling one to derive solutions of difference equations without making any special lucky guesses. Hydon emphasized on second-order difference equations, although his algorithm is valid for any order. When it comes to higher-order equations, computations can be cumbersome and extra ansatz may be needed to ease the calculations.

We aim to extend the work by Elsayed [1] where the author investigated the dynamics and solutions of

[TABLE]

where the initial conditions $x_{-5},\,x_{-4},\,x_{-3},\,x_{-2},\,x_{-1}$ and $x_{0}$ are arbitrary non-zero real numbers. For related work, see [2, 3]. One can notice that equations (1) are just special cases of a more general form

[TABLE]

where $(a_{n})_{n\in\mathbb{N}_{0}}$ and $(b_{n})_{n\in\mathbb{N}_{0}}$ are arbitrary sequences. We will use a symmetry based method to solve (2). Equivalently, we study

[TABLE]

instead, where $(A_{n})_{n\in\mathbb{N}_{0}}$ and $(B_{n})_{n\in\mathbb{N}_{0}}$ are arbitrary sequences. This means we can only compare $x_{i}$ with $u_{i+5}$. Furthermore, we use the same technique to obtain exact solutions of

[TABLE]

where $\lambda,\mu\in\mathbb{R}$, and again we study

[TABLE]

instead. Note that solutions of (4) were found in [8]; however, their method is completely different from ours.

2 Definitions and notation

The definitions are taken from Hydon [4] and most of the notation follows from the same book.

Definition 1

A parameterized set of point transformations,

[TABLE]

where $x=x_{i},$ $i=1,\dots,p$ are continuous variables, is a one-parameter local Lie group of transformations if the following conditions are satisfied:

$\Gamma_{0}$* is the identity map if $\hat{x}=x$ when $\varepsilon=0$* 2. 2.

$\Gamma_{a}\Gamma_{b}=\Gamma_{a+b}$* for every $a$ and $b$ sufficiently close to 0* 3. 3.

Each $\hat{x_{i}}$ can be represented as a Taylor series (in a neighborhood of $\varepsilon=0$ that is determined by $x$), and therefore

[TABLE]

Consider the $p$th-order difference equation

[TABLE]

for some function $\Omega$. Assume the point transformations are of the form

[TABLE]

with the corresponding infinitesimal symmetry generator

[TABLE]

where $S$ is the shift operator, i.e., $S:n\mapsto n+1$. The symmetry condition is defined as

[TABLE]

whenever (8) is true. Substituting the Lie point symmetries (9) into the symmetry condition (11) leads to the linearized symmetry condition

[TABLE]

whenever (8) holds.

Definition 2

$V_{n}$* is invariant under the Lie group of transformations (9) if and only if $XV_{n}=0$.*

We define the functions

[TABLE]

and we adopt the standard conventions

[TABLE]

We refer the reader to [4] for a deeper understanding of the concept of symmetry analysis of difference equations.

3 Main results

3.1 On the difference equations (3)

Consider the sixth-order difference equations of the form (3), that is,

[TABLE]

We impose the symmetry condition (12) and simplify the resulting equation to get

[TABLE]

To solve for $Q$, we first differentiate (16) with respect to $u_{n}$ ( keeping $\Omega$ fixed and viewing $u_{n+2}$ as a function of $u_{n},u_{n+4}$ and $\Omega$). This leads, after simplification, to

[TABLE]

where ′ denotes the derivative with respect to the independent variable. We then differentiate (17) with respect to $u_{n}$ to get

[TABLE]

The solutions of (18) are given by

[TABLE]

for some functions $\alpha_{n}$ and $\beta_{n}$ of $n$. To obtain more information on $\alpha_{n}$ and $\beta_{n}$, we substitute (19) in (16) and we split the resulting equation to get the following:

[TABLE]

These equations (20) reduce to

[TABLE]

The expression of $\beta_{n}$ in (21) is merely obtained by solving the corresponding characteristic equation $r^{4}+r^{2}+1=0$ for $r$. Assuming that $r_{i},\;i=1,2,3,4$, are the solutions of this characteristic equation, then $\beta_{n}$ is a linear combination of the $r_{i}^{n}$’s. In other words, the solutions of (21) are

[TABLE]

for some arbitrary constants $c_{i},\;i=1,\dots,4$, and where $\beta=\exp{\left({\pi i}/{3}\right)}$. Thus, we obtain four characteristics given by

[TABLE]

The four corresponding symmetry generators $X_{1},\,X_{2},\,X_{3}$ and $X_{4}$ are given by

[TABLE]

Here, using $Q_{4}$, we introduce the canonical coordinate [5]

[TABLE]

In view of (21), we introduce the variable

[TABLE]

It is easy to check that

[TABLE]

Therefore $r_{n}$ is invariant under $X_{4}$. It is advantageous to use

[TABLE]

that is, $\tilde{r}_{n}=\pm 1/(u_{n}u_{n+2}u_{n+4})$. Here, we choose to use the plus sign and we have shown, using (3), that

[TABLE]

Therefore,

[TABLE]

It is worthwhile to mention that equations in (30) give the solution of (29) for all $n$. By reversing all the change of variables, we have

[TABLE]

where $\tilde{r}_{k}$ is given in (30), $\Gamma_{n}=\exp\{\left(-\beta\right)^{n}c_{5}+\beta^{n}c_{6}+\bar{\beta}^{n}c_{7}+\left(-\bar{\beta}\right)^{n}c_{8}\}$ and $\gamma(l,k)=\beta^{l}\bar{\beta}^{k}$.

Note. Equation (31) gives the solution of (3) in a unified manner.

We can simplify (31) further by splitting it into six categories. We have

[TABLE]

Similarly, after a straightforward but lengthy computation, we get

[TABLE]

Note. We can obtain (34) using (28) which need not the use of absolute values.

Now, using (30) and (34), we obtain the solutions of (3) as follows:

[TABLE]

The implication is that solutions of (2) are given by

[TABLE]

3.2 The case where $(a_{n})$ and $(b_{n})$ are two-periodic sequences

Let $a_{n}=(\lambda,\mu,\lambda,\mu,\lambda,\dots)$, $b_{n}=(\eta,\zeta,\eta,\zeta,\dots)$, $\Phi=x_{-5}x_{-3}x_{-1}$ and $\Psi=x_{-4}x_{-2}x_{0}$. The solution in this case is given by the equations

[TABLE]

where $\left(\sum\limits_{l=0}^{s-1}\lambda^{l}\right)\eta\Phi\neq-\lambda^{s}$ and $\left(\sum\limits_{l=0}^{s-1}\lambda^{l}\right)\eta\Psi\neq-\lambda^{s},\quad s\leq 3n$.

3.3 The case where $(a_{n})$ and $(b_{n})$ are constants

Here, $a_{n}=\mu=\lambda$, $b_{n}=\eta=\zeta$, $\Phi=x_{-5}x_{-3}x_{-1}$ and $\Psi=x_{-4}x_{-2}x_{0}$. Equation (2) becomes $x_{n+1}={x_{n-5}}/{(\lambda+\eta x_{n-1}x_{n-3}x_{n-5})}$.

3.3.1 $\lambda=1$

The solution given in (36) simplifies to

[TABLE]

• •

If $\eta=1$, then equations in (38) are exactly the ones obtained in Theorem 2.1 in [1] for

[TABLE]

and their restriction (the initial conditions are arbitrary nonzero positive real numbers) is a special case of our restriction (the initial conditions are arbitrary nonzero real numbers and $s\Phi\neq-1,\,s\Psi\neq-1,\,0\leq s\leq 3n$ ).

• •

If $\eta=-1$, then equations in (38) are exactly the ones obtained in Theorem 3.1 in [1] for

[TABLE]

and their restriction (the initial conditions are arbitrary nonzero real numbers and $jbdf\neq 1,\,jace\neq 1$ ) coincides with our restriction (the initial conditions are arbitrary nonzero real numbers and $s\Phi\neq 1,\,s\Psi\neq 1,\,0\leq s\leq 3n$ ).

3.3.2 The case where $\lambda\neq 1$

In this case, the solution given in (36) simplifies to

[TABLE]

[TABLE]

[TABLE]

where $(1-\lambda^{s})\eta\Phi\neq-\lambda^{s}(1-\lambda)$ and $(1-\lambda^{s})\zeta\Psi\neq-\lambda^{s}(1-\lambda),\quad s\leq 3n$.

Note. If $\lambda=-1$ then the solution given in (3.3.2) simplifies to

[TABLE]

• •

When setting $\eta=1$ in (3.3.2), we get the result obtained in Theorem 4.1 in [1] for

[TABLE]

and their restriction coincides with our restriction (the initial conditions are arbitrary nonzero real numbers, $\Phi\neq 1$ and $\Psi\neq 1)$.

• •

When setting $\eta=-1$ in (3.3.2), we get the result obtained in Theorem 5.1 in [1] for

[TABLE]

and their restriction coincides with our restriction (the initial conditions are arbitrary nonzero real numbers, $\Phi\neq-1$ and $\Psi\neq-1)$.

4 On the difference equations (5)

Consider the fifth-order difference equations of the form (5), i.e.,

[TABLE]

Here, the procedure for finding the characteristics of (5) is similar as above and is as follows:

Impose the symmetry condition (12) to (5).

• -

Differentiate with respect to $u_{n}$ ( keeping $\Omega$ fixed) and viewing $u_{n+3}$ as a function of $u_{n},u_{n+1},\;u_{n+2}$ and $\Omega.$

• -

Differentiate with respect to $u_{n}$ twice (keeping $u_{n+1}$ fixed).

• -

Use the method of separation.

After preforming this series of operations, we obtain the characteristics

[TABLE]

where $\beta=\exp{\left(\frac{-2\pi i}{3}\right)}$. Thus, we obtain two characteristics with corresponding generators given by

[TABLE]

From the characteristic equations

[TABLE]

we obtain the invariants $c_{1}={u_{n+1}^{\beta}}/{u_{n}},c_{2}={u_{n+2}^{\bar{\beta}}}/{u_{n}},c_{3}=u_{n}u_{n+3},c_{4}={u_{n+4}^{\beta}}/{u_{n}}$ and $c_{5}=V_{n}.$ We readily notice that

[TABLE]

and we choose $V_{n}=c_{1}^{\bar{\beta}}c_{2}^{\beta}$, i.e.,

[TABLE]

By shifting (49) thrice, we get

[TABLE]

whose solution is given by

[TABLE]

The constants $c_{i},\;i=6,\dots,10$, can be obtained from the following equations:

[TABLE]

Thanks to (49), we can express $u_{n}$ in terms of $V_{n}$ as follows:

[TABLE]

Equations in (53) give the solutions of (5) in a unified manner.

For the sake of clarification, we split solutions (53a) to realise the solutions in the existing literature. Using (53a) and (53b) , we have

[TABLE]

Using the same approach, we have shown that

[TABLE]

4.1 The case of ${\lambda}=1$

Equation (5) becomes

[TABLE]

and we said earlier that the solution of (50), in this case, is (51), i.e.,

[TABLE]

We have

[TABLE]

and using (52a) in (58), we get

[TABLE]

Using the same approach, we have shown that

[TABLE]

Let $a=x_{-4},\;b=x_{-3},\;c=x_{-2},\;d=x_{-1}$ and $A=u_{0},\;B=u_{1}$, $C=u_{2},$ $D=u_{3}$ and $E=u_{4}$. Using (60) in (55), we obtain the solution of (56) as follows:

[TABLE]

whenever the denominators do not vanish.

4.2 The case of ${\lambda}\neq 1$

In this case, as we found earlier, the solution of (50) is given by (51), i.e.,

[TABLE]

Using this, we get

[TABLE]

and using (52d) in (63), we find

[TABLE]

Using the same approach, we have shown that

[TABLE]

Using (65) in (55), we obtain the solution of (5) as follows:

[TABLE]

[TABLE]

[TABLE]

where $\Delta_{i}=u_{i}u_{i+1}u_{i+2}$. Equations in (4.2) give the exact solution of (5) for any real values of $\lambda$ and $\mu$ provided that the denominators do not vanish.

Recall that we acted the shift operator on (4) to get (5). Hence, the solutions of (4) are obtained, using (4.2), as follows:

[TABLE]

[TABLE]

[TABLE]

for any real values of $\lambda$ and $\mu$ as long as the denominators do not vanish.

• •

When $\lambda=1$ and ${\mu}=1$, equations in (4.2) yield the results obtained by Yasin Yazlik in Theorem 5 in [9] for

[TABLE]

where $a,\;b,\;c,\;c,\;d$ and $e$ are positive real numbers.

• •

When $\lambda=1$ and ${\mu}=-1$, equations in (4.2) yield the results obtained by Yasin Yazlik in Theorem 9 in [9] for

[TABLE]

where $a,\;b,\;c,\;c,\;d$ and $e$ are positive real numbers with ${a}{b}{c}\neq 1$ and ${c}{d}{e}\neq 1$ .

**Note. **There should not be a minus sign right after the expression of $x_{3n-2}$ in Theorem 9 in [9].

• •

When $\lambda=-1$ and $\mu=1$, equations in (4.2) yield the results obtained by Yasin Yazlik in Theorem 7 in [9] for

[TABLE]

where $a,\;b,\;c,\;d$ and $e$ are non zero real numbers with $abc\neq 1$,  $bcd\neq 1$ and $cde\neq 1$.

• •

When $\lambda=-1$ and $\mu=-1$, equations in (4.2) yield the results obtained by Yasin Yazlik in Theorem 11 in [9] for

[TABLE]

where $a,\;b,\;c,\;c,\;d$ and $e$ are non zero real numbers with ${a}{b}{c}\neq-1$ , ${b}{c}{d}\neq-1$ and ${c}{d}{e}\neq-1$ .

**Note. **There should be a minus sign right after the expression of $x_{6n+1}$ in Theorem 11 in [9].

5 Conclusion

In this paper, we have obtained nontrivial symmetries of some rational ordinary difference equations and their exact solutions were obtained. Most importantly, we note that (21) gives a clear idea, without making any lucky guesses, for the most convenient choice of the invariant of (3).

Conflict of interests

The authors declare that there is no conflict of interest regarding the publication of this paper.

Data availability statement

No data were used to support this study.

Funding statement

The authors received no specific funding for this work.