Abstract
D.Sc. (Mathematics)
In this thesis aspects of continuous symmetries of differential equations are studied. In
particular the following aspects are studied in detail: Lie algebras, the Lie derivative, the
jet bundle formalism for differential equations, Lie point and Lie-Backlund symmetry
vector fields, recursion operators, conservation laws, Lax pairs, the Painlcve test, Lie
algebra valued differenmtial forms and Dose operators as a representation of differential
operators.
The purpose of the study is to gain a better understanding of complicated nonlinear
dirrerential equations that describe nature and to construct solutions. The differential
equations under consideration were derived [rom physics and engineering. They are
the following: the Kortcweg-dc Vries equation, Burgers' cquation , the sine-Gordon equation,
nonlinear diffusion equations, the Klein Gordon equation, the Schrodinger equation,
nonlinear Dirac equations, Yang-Mills equations, the Lorentz model, the Lotka-Volterra
model, damped unharrnonic oscillators, and others.
The newly found results and insights are discussed in chapters 8 to 17. Details on the
COli tents of each chapter and rcfernces to some of my articles arc given in chapter 1.