The Bäcklund transformation and nonlinear superposition formula of solutions for the Liouville's equation in higher dimensions
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In this paper, we show that Bäcklund transformation derived by Leibbrandt et al. for the Liouville's equation in three spatial dimensions,\(\nabla ^2 \alpha = \exp \alpha ,\nabla ^2 \partial _k^2 + \partial ^2 + \partial _r^2 \) can be decomposed into several Bäcklund transformations for the same equation in two spatial dimensions, moreover, the superposition formula which is derived from this transformation is actually invalid, thus the discussions based on that formula is incorrect as well. We also considered some results about the Liouville's equation in N spatial dimensions.
KeywordsMathematical Modeling Industrial Mathematic High Dimension Spatial Dimension Nonlinear Superposition
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