Geometrical interpretations of Bäcklund transformations and certain types of partial differential equations : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University

Abstract

Gauss' Theorema Egregium contains a partial differential equation relating the Gaussian curvature K to components of the metric tensor and its derivatives. Well known partial differential equations such as the Schrödinger equation and the sine-Gordon equation correspond to this PDE for special choices of K and special coördinate systems. The sine-Gordon equation, for example, can be derived via Gauss' equation for K = –1 using the Tchebychef net as a coördinate system. In this thesis we consider a special class of Bäcklund Transformations which correspond to coördinate transformations on surfaces having a specified Gaussian curvature. These transformations lead to Gauss' PDE in different forms and provide a method for solving certain classes of non-linear second order partial differential equations. In addition, we develop a more systematic way to obtain a coordinate system for a more general class of PDE, such that this PDE corresponds to the Gauss equation.