Abstract
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.
Highlights
The role and methods associated with conservation laws are 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
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
The following proposition that defines the relationship between point symmetries, multipliers, and conservation laws constructed via the Noether operator in [11] can be proved
Summary
The role and methods associated with conservation laws are 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. 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, the Euler operator annihilates this product so that finding conserved flows amounts to finding the factors. 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. 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
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