Symmetries, conservation laws, and ‘integrability’ of difference equations
© Folly-Gbetoula and Kara; licensee Springer. 2014
Received: 10 March 2014
Accepted: 29 July 2014
Published: 15 August 2014
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.
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  and [9, 10], inter alia. The method for the construction of the conservation laws employed here follows that introduced in . The variational approach, not followed here, uses the equivalent of Noether symmetries and can be found in .
We regard the domain of a given partial differential equation (PDE) as a fiber bundle , 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 . 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 . 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 , 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
In order to find F and G, we consider the theory provided by Hydon in . 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
where ω is the right-hand side of the discrete Liouville (6).
where is a function of k and l. Note that 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 (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 ).
Here, 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  for the symmetry underlying relation that exists. Very recently, this idea has been initiated and discussed for PDEs in . 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
4.1 The discrete Liouville equation
4.2 The discrete sine-Gordon equation
5 Concluding remarks
By replacing in (94) the function with its expression given by (52), we obtained the constraint (7), i.e., . 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  using the singularity confinement condition. It is also precisely the one obtained in  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.
MKF-G and AHK would like to thank the referees of this manuscript for their valuable comments and suggestions.
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