On the relationship between the invariance and conservation laws of differential equations

In this paper, we highlight the complimentary nature of the results of Anco&Bluman and Ibragimov in the construction of conservation laws; that whilst the former establishes the role of multipliers, the latter presents a formal procedure to determine the flows. Secondly, we show that there is an underlying relationship between the symmetries and conservation laws in a general setting - extending the results of Kara&Mahomed. The results take apparently differently forms for point symmetry generators and higher-order symmetries. Similarities exist, to some extent, with a previously established result relating symmetries and multipliers of a differential equation. A number of examples are presented.


Introduction
The role and methods associated with conservation laws are now well established and there have been some momentous works in these areas in recent times building on the contributions made by Noether which generally dealt with variational problems those that admit variational symmetries. It is not surprising then that much of the recent works focused on generalisations as far as constructions of conservation laws go, possibly nonvariational and preferably independent of a knowledge of symmetries. A vast amount and extensively cited works are due to Anco & Bluman in [1,2], inter alia, Anderson [3,4], and Kara & Mahomed [5], and a useful indepth treatise is presented in the work of Olver [6] which goes a long way in discussing the concept of "recursion operators". The first of these deals extensively with the notion of "multipliers" that if a differential equation times a factor (differential function) is a total divergence, then the Euler operator annihilates this product so that finding conserved flows amounts to finding the factors. It turns out that the multipliers are solutions of the adjoint equation. Of course, one still needs to determine the corresponding conserved flows using, amongst others, homotopy formulae [7]. A large amount of software to construct the various components of conserved vectors is available; see [8,9].
Since conservation laws seem to be tied in with invariance properties, the intention to avoid the symmetry route can prove to be difficult. This is partly due to the amount of work required to construct conserved flows; it can be cumbersome and tedious when dealing with the large systems of differential equations that arise in physics, cosmology, and engineering. For example, constructing conservation directly from the definition may be straightforward for simple scalar ordinary differential equations but the more complex the differential equation, as they are in fluids, cosmology, and the various systems of Schrödinger equations that is abundant in the literature (to name a few), the greater the task. The popularity of Noether's theorem lies in the existence of a formula. Trying to mimic this formula even in the nonvariational case has been tempting and partly successful; see [10]. In particular, the recent work of Ibragimov [11] develops a procedure to construct conserved vectors using the Noether operator, a symmetry of the differential equation solutions of the adjoint equation.
An in-depth study into the results due to Anco & Bluman in [1,2] and Ibragimov [11] suggests that similarities are abundant; see [12]. However, it also shows that since the methods employed are largely different, there are some intrinsic differences and what is presented here is an attempt to show that these differences, in fact, allow these works to 2 Journal of Mathematics complement each other. For example, the underlying aspect in the multiplier approach is primarily to construct multipliers that leads to the differential equation being conserved. These multipliers can be chosen with a specific order (in derivatives) in mind and then one may choose from a number of methods to construct the conserved vectors. In [11], the particular method appeals to the Noether operator after having knowledge of a symmetry and a solution of the adjoint equation. It will be shown that the total divergence of the conserved flow has a form dependent on whether the symmetry used is a point symmetry or an evolutionary/canonical symmetry; the general result in the latter case would include generalised symmetries.

Notations and Preliminaries
What follows is a summary of the definitions, concepts, and notations that will be utilised in the sequel.
Conservation laws may be expressed as conserved forms [4]. For example, if x = ( , ), the conserved form would be (where ( , ) is the conserved vector such that + = 0 on the solutions of the pde ( , , , (1) , . . . , ( ) ) = 0 ). Here, d leads to the "conserved density" if and are time and space, respectively.

Conservation Laws
. . In the first case, we consider the relationship between the conserved flows and the respective point symmetry generators of the differential equation.
The total divergence is That is, the total divergence takes the form so that where Y = + .
The following proposition that defines the relationship between point symmetries, multipliers, and conservation laws constructed via the Noether operator in [11] can be easily proved.  After some cumbersome calculations, Proposition 3 is easily generalised to the multidimensional pde ( , , , ( ) , , . . . , ( ) ) = 0.
. . We now consider the connection between generalised symmetries, higher-order symmetries, and evolutionary/canonical symmetries and associated conservation laws. Again, we suppose L = V( , ) [11].
(i) With 1 , we obtain the components of the conserved vector to be so that where = + (1/2) and R 1 = + (1/2) are the recursion operator associated with 1 .
(ii) Using 2 , we get so that (iii) With 3 , we get so that after some simplifications the total divergence is where = 2 + and R 3 = + 2 are the respective recursion operator.
Example . It is well known that, for the wave equation − = 0 and any variational equation, "multipliers" or, equivalently, solutions of the adjoint equation are symmetries of the equation so that in the simple case of the evolutionary vector field = ( + ) is a generalised symmetry and = + is multiplier. Applying Noether's theorem is clearly the efficient route to construct a conservation law. Alternatively, if we assume = V( , )( − ) using the procedure in [11], we get so that where = + and R = + .
Again, the proposition can be generalised to the multidimensional case.

Discussion
It is clear that, in each case, the conserved flows ( , ) are nontrivial since + do not vanish identically but, rather, on the solutions of the differential equation. The dependence of this method on solutions of the adjoint equation is equivalent to the multiplier approach since multipliers are solutions of the adjoint equation. Thus, as mentioned before, the two approaches in [1,11] complement each other and the latter has a formal procedure to construct the conserved flows using symmetries of the differential equation. Moreover, we showed that the total divergence, quite explicitly, displays a relationship between symmetries (point or generalised) and conservation laws in a general setting; compare this to the results in [5]. Also, the main results of this paper mimic, to some extent, the results established on the relationship between symmetries and multipliers of a differential equation as discussed in [14].

Data Availability
No data were used to support this study.

Conflicts of Interest
The author declares that they have no conflicts of interest.