关于高维Liouville方程的Bäcklund变换和解的非线性叠加公式
The Backlund Transformation and Nonlinear Superposition Formula of Solutions for the Liouville’s Equation in Higher Dimensions
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摘要: 本文指出了由Leibbrandt等人导出的关于三维空间Liouville方程∇2a=expa∇2=∂x2+∂y2+∂z2,的Bäcklund变换可以分解成几个二维空间Liouville方程的Backlund变换的组合 同时,由该变换导出的解的非线性叠加公式实际上是无效的, 从而一些基于这一公式的讨论也不正确.文中还考虑了有关N维空间Liouville方程的一些结果.Abstract: In this paper, we show that Backlund transformation derived by Leibbraadt et al for the Liouvilles equation in three spatial dimeasions,∇2a=expa∇2=∂x2+∂y2+∂z2 can be decomposed into several Backlund 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.
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[1] Solitons 82,Abstracts of Cpaference aced Workshop Talks and.Posters.Edinburgh,England,Aug,(1982). [2] Leibbrandt,G.,S,S,Wang and N,Zamani,Backlund generated solutions of Liouville's equdtion and their graphical representations in three spatial dimensioas,d.Math.Phys.,23,9(1982),1566-1572. [3] Leibbrandt,G.,Nonlinear superposition for Liouville's equation in three spatial dimensions,Left,Math,Phys.,4(1984).317-321. -
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