Applied Mathematics and Mechanics

, Volume 5, Issue 6, pp 1777–1782 | Cite as

The Bäcklund transformation and nonlinear superposition formula of solutions for the Liouville's equation in higher dimensions

  • Huang Xun-Cheng


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.


Mathematical Modeling Industrial Mathematic High Dimension Spatial Dimension Nonlinear Superposition 
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    Soltons' 02, Abstracts of Conference and Workshpo Talls and Posters, Edinburge, England, August (1982).Google Scholar
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    Leibbrandt G., S.S. Wang and N. Zamani, Eäcklund generated solutions of Liouville's equation and their graphical representations in three spatial dimensions, J. Math. Phys., 23, 9(1982), 1566–1572.CrossRefGoogle Scholar
  3. (3).
    Leibbrandt G., Nonlinear superposition for Liouville's equation in three spatial dimensions, Lett. Math. Phys., 4(1980), 317–321.CrossRefGoogle Scholar

Copyright information

© HUST Press 1984

Authors and Affiliations

  • Huang Xun-Cheng
    • 1
  1. 1.Shanghai Institute of Computing TechniqueShanghai

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