In the first part of this work, two nonlinear partial differential equations, namely, a modified Camassa-Holm-Degasperis-Procesi equation and the generalized Kortewegde Vries equation with two power law nonlinearities are studied. The Lie symmetry method along with the simplest equation method is used to construct exact Solutions for these two equations. The second part looks at two systems of partial differential equations, namely, the generalized Boussinesq-Burgers equations and the (2+1)-dimensional Davey-Stewartson equations. The Lie symmetry method and the travelling wave hypothesis approach are utilized to obtain exact solutions of the generalized Boussinesq-Burgers equations. The travelling wave hypothesis approach is used to find exact solutions of the (2+ 1 )-dimensional Davey-Stewartson equations. / Thesis (Msc. in Applied Mathematics) North-West University, Mafikeng Campus, 2012

Thesis (doctoral)--Georg-August-Universität zu Göttingen, 1912. / Cover title. Vita. Includes bibliographical references.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1897. / Vita. Includes bibliographical references.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1890. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of Chicago. / Vita. "Private edition distributed by the University of Chicago libraries, Chicago, Illinois."

Thesis (Ph. D.)--University of Pennsylvania, 1911. / Vita.

The objective of this thesis is to study meromorphic solutions of complex algebraic ordinary differential equations (ODEs). The thesis consists of two main themes. One of them is to find explicitly all meromorphic solutions of certain class of complex algebraic ODEs. Since constructing explicit solutions of complex ODEs in general is very difficult, the other theme (motivated by the classical conjecture proposed by Hayman in 1996) is to establish estimations on the growth of meromorphic solutions in terms of Nevanlinna characteristic function. The tools from complex analysis that will be used have been collected in Chapter 1. Chapter 2 is devoted to introducing a method, which was first used by Eremenko and later refined by Conte and Ng, to give a classification of some complex algebraic autonomous ODEs. Under certain assumptions, based on local singularity analysis and Nevanlinna theory, this method shows that all meromorphic solutions of these ODEs if exist, must belong to ‘class W’, which consists of elliptic functions and their degenerations. Combined with knowledge from function theory, as shown by Demina and Kudryashov, it further allows us to find all of them explicitly and the details of the method will be illustrated by constructing new real meromorphic solutions of the stationary case of cubic-quintic Swift-Hohenberg equation. In Chapter 3, the same method is used to construct on R^n, n ≥ 2 some explicit Bryant solitons and on R^n\{0}, n ≥ 2 some Ricci solitons, and one of them turns out to be a new Ricci soliton on R^5\{0}. In addition, the completeness of corresponding metrics on the Ricci solitons that we have constructed are also discussed. In 1996, Hayman conjectured an upper bound on the growth, in terms of Nevanlinna characteristic function, of meromorphic solutions of complex algebraic ODEs. Related work in the literature towards this so-called classical conjecture is first reviewed in Chapter 4. The classical conjecture for three types of second order complex algebraic ODEs will then be verified by either giving a classification of the meromorphic solutions or obtaining them explicitly in Chapter 4. As the classical conjecture seems to be out of reach at present, we proposed in Chapter 5 to study a particular class of complex algebraic ODEs which can be factorized into certain form. On one hand, for these factorizable ODEs, it has been proven for the generic case that all their meromorphic solutions must be elliptic functions or their degenerations. On the other hand, the second order factorizable ODEs have been carefully studied so that their meromorphic solutions have been obtained explicitly except one case. This will allow the classical conjecture for most of the second order factorizable ODEs to be verified by employing Nevanlinna theory and certain qualitative results from complex differential equations. Finally, the classical conjecture has been shown to be sharp in certain cases. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

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