Annals of Global Analysis and Geometry

, Volume 11, Issue 1, pp 49–64 | Cite as

Weingarten surfaces and nonlinear partial differential equations

  • Hongyou Wu
Article

Abstract

The sine-Gordon equation has been known for a long time as the equation satisfied by the angle between the two asymptotic lines on a surface inR3 with constant Gauss curvature −1. In this paper, we consider the following question: Does any other soliton equation have a similar geometric interpretation? A method for finding all the equations that have such an interpretation using Weingarten surfaces inR3 is given. It is proved that the sine-Gordon equation is the only partial differential equation describing a class of Weingarten surfaces inR3 and having a geometricso(3)-scattering system. Moreover, it is shown that the elliptic Liouville equation and the elliptic sinh-Gordon equation are the only partial differential equations describing classes of Weingarten surfaces inR3 and having geometricso(3,C)-scattering systems.

Key words

Weingarten surfaces soliton equations geometric scattering systems 

MSC 1991

Primary 53A05 35Q53 Secondary 53C21 

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Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Hongyou Wu
    • 1
  1. 1.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

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