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On the solution of a class of second-order quasi-linear PDEs and the Gauss equation

Published online by Cambridge University Press:  17 February 2009

A. R. Selvaratnam ,
M. Vlieg-Hulstman ,
B. van-Brunt  and
W. D. Halford
A. R. Selvaratnam
Affiliation:
Department of Mathematics, Massey University, Palmerston North, New Zealand.
M. Vlieg-Hulstman
Affiliation:
Department of Mathematics, Massey University, Palmerston North, New Zealand.
B. van-Brunt
Affiliation:
Department of Mathematics, Massey University, Palmerston North, New Zealand.
W. D. Halford
Affiliation:
Department of Mathematics, Massey University, Palmerston North, New Zealand.
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Abstract

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Gauss' Theorema Egregium produces a partial differential equation which relates the Gaussian curvature K to components of the metric tensor and its derivatives. Well-known partial differential equations (PDEs) such as the Schrödinger equation and the sine-Gordon equation can be derived from Gauss' equation for specific choices of K and coördinate systems. In this paper we consider a class of Bäcklund Transformations which corresponds to coördinate transformations on surfaces with a given Gaussian curvature. These Bäcklund Transformations lead to the construction of solutions to certain classes of non-linear second order PDEs of hyperbolic type by identifying these PDEs as the Gauss equation in some coördinate system. The possibility of solving the Cauchy Problem has also been explored for these classes of equations.

Type
Research Article
Information
The ANZIAM Journal , Volume 42 , Issue 3 , January 2001 , pp. 312 - 323
Copyright
Copyright © Australian Mathematical Society 2001

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