Group analysis of differential equations: A new type of Lie symmetries

摘要

We construct a new type of symmetries using the regular Lie symmetries as the basis, which we call Modified symmetries. The contrast is that while Lie symmetries arise from point transformations, the Modified symmetries result from the transformations of the neighborhood of that point. The similarity is that as the neighborhood contracts to the central point, the two sets of symmetries become indistinguishable from one another, meaning the Modified symmetries will cease to exist if there were no Lie symmetries in the first place. The advantage is that the group invariant solutions are not affected by all these, because they result from ratios of the symmetries, and will therefore exist in the absence of Lie symmetries, i.e. zero symmetries. Zero symmetries lead to 0/0, and no further. With the Modified symmetries we get f(x, ω)/g(x, ω) = 0/0 as ω goes to zero, and there are numerous mathematical techniques through which this can be resolved. We develop this concept using tensors and exterior calculus, and elaborate its application exhaustedly through a numerical example.