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# Symmetries, conservation laws, and ‘integrability’ of difference equations

- Mensah K Folly-Gbetoula
^{1}and - Abdul H Kara
^{1, 2}Email author

**2014**:224

https://doi.org/10.1186/1687-1847-2014-224

© Folly-Gbetoula and Kara; licensee Springer. 2014

**Received:**10 March 2014**Accepted:**29 July 2014**Published:**15 August 2014

## Abstract

A number of nontrivial conservation laws of some difference equations, *viz.*, the discrete Liouville equation and the discrete sine-Gordon equation, are constructed using first principles. Symmetries and the more recent ideas and notions of characteristics (multipliers) for difference equations are also discussed.

## Keywords

- conservation laws
- symmetries
- difference equations
- discrete Liouville equation
- discrete sine-Gordon equation

## 1 Introduction

The role of symmetries of difference equations is now well established and the applications of the symmetries in the analysis (especially reduction) of the equations are also well documented (see [1–4]). However, the role and construction of conservation laws for partial difference equations (PDEs), to the best of our knowledge, is somewhat new but the preliminary concepts and definitions are available even in the context of variational equations (see [5–7]). These conservation laws, as in the case of differential equations, have a variety of applications especially as another tool in the reduction of the equation under scrutiny.

The aim of this work is to obtain the conservation laws of PDEs which are of interest, *viz.*, the discrete Liouville equation and the discrete sine-Gordon equation. These equations were studied in [8] and [9, 10], *inter alia*. The method for the construction of the conservation laws employed here follows that introduced in [5]. The variational approach, not followed here, uses the equivalent of Noether symmetries and can be found in [7].

We regard the domain of a given partial differential equation (PDE) as a fiber bundle $M=X\times U$, where *X* is the base space of independent variables and *U* is the vertical space, *i.e.*, the fiber of dependent variables **u** over each $\mathbf{x}\in X$. The direct method for constructing conservation laws of PDEs requires the domain *M* to be topologically trivial, which occurs if each fiber *U* and the base space *X* are star-shaped (see Poincaré’s lemma).

Symmetries and conservation laws are useful tools for finding exact solutions to differential equations. The association of symmetries, conservation laws and integrability was established for differential equations [11]. It has been shown that when the symmetry generator and the first integral (conservation laws) are associated via the invariance condition, one may proceed to double reduction of the equation. Consequently, these properties should be retained when discrete analogs of such equations are constructed. Several methods for obtaining a discretization of a differential equation have been studied (see [12–15]).

As far as PDEs are concerned, we can write the domain as $M=X\times U$, but now *X* is the set of integer-valued multi-indices **n** that label each lattice point (we assume that the lattice points are labeled sequentially, without jumps; this does not require the lattice to be uniform).

## 2 Conservation laws of the discrete Liouville equation and the discrete sine-Gordon equation

*k*and

*l*are integers, ${u}_{k}^{l}$ is a function that depends on the independent variables

*k*and

*l*,

*ω*is a function of the dependent and independent variables. Finding the conservation laws requires the knowledge of the shift operators. They are defined as follows:

*F*and

*G*are functions of the dependent and independent variables, id is the identity mapping. We are looking for conservation laws that lie on the quad-graph and we are interested in finding nontrivial conservation laws. We then assume that the functions

*F*and

*G*are of the form

In order to find *F* and *G*, we consider the theory provided by Hydon in [5]. In his paper, Hydon applied the method to scalar partial difference equations that are second order in one variable but in this paper we are dealing with first order difference equations in two variables.

### 2.1 The discrete Liouville equation

*k*and

*l*. Note that (3) can be written as follows:

where *ω* is the right-hand side of the discrete Liouville (6).

*F*and

*G*are solutions to (8), then $F+{F}_{1}(k,l)$ and $G+{G}_{1}(k,l)$ are also solutions for any functions ${F}_{1}$ and ${G}_{1}$. One can see from (8) that

*F*and

*G*take different arguments. To overcome this, we eliminate terms that depend on

*ω*, by differentiating with respect to ${u}_{k}^{l+1}$ and ${u}_{k+1}^{l}$, respectively, keeping omega fixed. The derivative of (8) with respect to ${u}_{k+1}^{l}$ is given by

*G*we differentiate two times with respect to ${u}_{k}^{l+1}$. This yields

*F*does not depend on ${u}_{k}^{l+1}$, therefore we can separate by powers of ${u}_{k}^{l+1}$ to get

*G*only. The first derivative with respect to ${u}_{k+1}^{l}$ gives

*G*does not depend on ${u}_{k+1}^{l}$. This allows us to equate the coefficients of powers of ${u}_{l}^{k+1}$ to zero. We get

*F*and

*G*, given by (17) and (22), into (14) and (19) gives

*k*and

*l*. The substitution of (27) and (28) in (11) did not provide any new information on the unknown functions. We then continue our computations by substituting (27) and (28) into (9), and separating the resulting equation by powers of ${u}_{k}^{l+1}$. This yields

where ${K}_{2}$ is a function of *k* and *l*. Note that $({F}_{2},{G}_{2})$ is a trivial one.

### 2.2 The discrete sine-Gordon equation

Equation (38) represents a nonautonomous extension of the lattice sine-Gordon equation. In the continuous limit, this nonautonomous form goes over to ${w}_{x,t}=f(x)g(t)sinw$ (this explicit *x* and *t* dependence can be absorbed through a redefinition of the independent variables leading to the standard, autonomous, sine-Gordon case, but no such gauge exists in the discrete case [8]).

*ω*is given by

*F*to obtain an equation that involves

*G*only. For the same reason as before, we equate coefficients of powers of ${u}_{k}^{l+1}$. The resulting system can be summarized as follows:

*F*and

*G*into the previous equations. After a set of long calculations, we find that the conservation laws for the discrete sine-Gordon equation are given by

Here, $({F}_{2},{G}_{2})$ is trivial.

## 3 On multipliers/characteristics

The relationship between the characteristics and conserved vectors of PDEs was well known for variational equations and was generalized relatively recently. In fact, the characteristics are the conserved vectors. The use of these in the construction of conservation laws has been discussed in detail in [16, 17] and [18] for the symmetry underlying relation that exists. Very recently, this idea has been initiated and discussed for PDEs in [19]. Below, we present the multipliers for the nontrivial cases of conservation laws that arise above.

### 3.1 The discrete Liouville equation

### 3.2 The discrete sine-Gordon equation

respectively.

## 4 Symmetries

*i.e.*,

*Q*is called the characteristic of the one-parameter group. We also have

*ϵ*yields the linearized symmetry condition

### 4.1 The discrete Liouville equation

*X*is the symmetry generator given by

*ω*fixed),

*i.e.*, we apply the differential operator ${L}_{0}=(\partial /\partial {u}_{k}^{l})+(\partial {u}_{k+1}^{l}/\partial {u}_{k}^{l})(\partial /\partial {u}_{k+1}^{l})$. This gives

*i.e.*,

*k*and

*l*. We then substitute (80) in (74) to get the following constraint:

### 4.2 The discrete sine-Gordon equation

*Q*after a series of steps. Firstly, we multiply (88) by ${({u}_{k}^{l})}^{2}$. Secondly, we differentiate the resulting equation with respect to ${u}_{k}^{l}$. Finally, we simplify to get the equation

*B*and

*C*are functions of

*k*and

*l*. The substitution of (90) in (89) and (87) puts more constraints on the unknown functions

*B*and

*C*. After a set of long calculations we get

## 5 Concluding remarks

*F*and

*G*satisfied. The unknown functions were naturally found by going up the hierarchy but surprisingly we were also able to find another constraint on

*z*. In fact the shift operator, ${S}_{l}$, acts on (50) to produce the following equation:

By replacing in (94) the function ${c}_{3}(k+1,l)$ with its expression given by (52), we obtained the constraint (7), *i.e.*, ${z}_{k+1}^{l+1}{z}_{k}^{l}-{z}_{k}^{l+1}{z}_{k+1}^{l}=0$. The constraint (7) turns out to be a sufficient condition for the discrete Liouville equation and the discrete sine-Gordon equation to have nontrivial conservation laws (note that the substitution of (35), (36), and (37) into (8) leads to the same constraint (7)). This condition was obtained in [20] using the singularity confinement condition. It is also precisely the one obtained in [8] using the techniques of the study of the degree of the iterate.

We note that the association of symmetries, conservation laws, and integrability for difference equations is as important and conclusive as was established for differential equations even in the non-variational case.

## Declarations

### Acknowledgements

MKF-G and AHK would like to thank the referees of this manuscript for their valuable comments and suggestions.

## Authors’ Affiliations

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