Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T06:11:01.980Z Has data issue: false hasContentIssue false

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
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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Copyright
Copyright © Australian Mathematical Society 2001

References

[1] Eisenhart, L. P., A Treatise on the Differential Geometry of Curves and Surfaces (Ginn, 1909) (reprint 1960).Google Scholar
[2] Garabedian, P. R., Partial Differential Equations (John Wiley and Sons, 1964).Google Scholar
[3] Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series and Products (Academic Press, 1980).Google Scholar
[4] Johnston, M., “Geometry and the sine-Gordon equation”, Technical Report AMR 95/16, University of New South Wales, Australia, 1995.Google Scholar
[5] Kevorkian, J., Partial Differential Equations, Analytical Solution Techniques (Chapman and Hall, 1993).Google Scholar
[6] Konopelchenko, B. G., “Solitons of curvature”, Acta Applicandae Mathematicae 39 (1995) 379387.CrossRefGoogle Scholar
[7] Lawden, D. F., Elliptic Functions and Applications (Springer-Verlag, 1989).CrossRefGoogle Scholar
[8] Marden, L., Calculus of Several Variables (George Allen and Unwin, 1971).Google Scholar
[9] Pogorelov, A. V., Differential Geometry (P. Noordhoff, Groningen, The Netherlands, 1959).Google Scholar
[10] Rogers, C. and Shadwick, W. F., Bäcklund Transformations and Their Applications (Academic Press, 1982).Google Scholar
[11] Stoker, J. J., Differential Geometry (John Wiley and Sons, 1989).Google Scholar
[12] Struik, D. J., Lectures on Classical Differential Geometry (Addison-Wesley, 1961).Google Scholar