In this work we study the applications of Lie symmetry analysis to certain nonlinear evolution equations of mathematical physics. Exact solutions and conservation laws are obtained for such equations. The equations which are considered in this thesis are a generalized Korteweg-de Vries-Burgers equation, a two-dimensional integrable generalization of the Kaup-Kupershmidt equation, a coupled Korteweg-de Vries system, a generalized coupled variable-coefficient modified Korteweg-de Vries system, a new coupled Korteweg-de Vries system and a new coupled Kadomtsev-Petviashvili system. The generalized Korteweg-de Vries-Burgers equation is investigated from the point of view of Lie group classification. We show that this equation admits a four-dimensional equivalence Lie algebra. It is also shown that the principal Lie algebra consists of a single translation symmetry. Several possible extensions of the principal Lie algebra are computed and their associated symmetry reductions and exact solutions are obtained. The Lie symmetry method is performed on a two-dimensional integrable generalization of the Kaup-Kupershmidt equation. Exact solutions are obtained using the Lie symmetry method in conjunction with the extended tanh method and the extended Jacobi elliptic function method. In addition to exact solutions we also present conservation laws which are derived using the multiplier approach. A coupled Korteweg-de Vries system and a generalized coupled variable-coefficient modified Korteweg-de Vries system are investigated using Lie symmetry analysis. The similarity reductions and exact solutions with the aid of simplest equations and Jacobi elliptic function methods are obtained for the coupled Korteweg-de Vries system and the generalized coupled variable-coefficient modified Korteweg-de Vries system. In addition to this, the conservation laws for the two systems are derived using the multiplier approach and the conservation theorem due to Ibragimov. Finally, a new coupled Korteweg-de Vries system and a new coupled Kadomtsev Petviashvili system are analyzed using Lie symmetry method. Exact solutions are obtained using the Lie symmetry method in conjunction with the simplest equation, Jacobi elliptic function and (G'/G)-expansion methods. Conservation laws are also obtained for both the systems by employing the multiplier approach. / Thesis (PhD.(Applied Mathematics) North-West University, Mafikeng Campus, 2013

Thesis (Ph.D.)--University of Toledo, 2006. / Typescript. "A dissertation [submitted] as partial fulfillment of the requirements of the Doctor of Philosophy degree in Mathematics." Bibliography: leaves 79-82.

No description available.

1 v. (various paging) ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.1980) from the Dept. of Mathematical Physics, University of Adelaide

Thesis (Ph.D. 1980) from the Department of Mathematical Physics, University of Adelaide.

Originally presented as author's thesis, Bonn, 1983. / Bibliography: p. 74.

No description available.

This research studies two nonlinear problems arising in mathematical physics. Firstly the Korteweg-de Vrics-Burgers equation is considered. Lie symmetry method is used to obtain t he exact solutions of Korteweg-de Vries-Burgers equation. Also conservation laws are obtained for this equation using the new conservation theorem. Secondly, we consider the generalized (2+ 1)-dimensional Zakharov-Kuznctsov (ZK) equation of time dependent variable coefficients from the Lie group-theoretic point of view. We classify the Lie point symmetry generators to obtain the optimal system of one-dimensional subalgebras of t he Lie symmetry algebras. These subalgebras arc then used to construct a number of symmetry reductions and exact group-invariant solutions of the ZK equation. We utilize the new conservation theorem to construct the conservation laws of t he ZK equation. / Thesis (M. Sci in Applied Mathematics) North-West University, Mafikeng Campus, 2011

The purpose of this thesis is to study those non-flat space-times in General Relativity admitting high dimensional Lie groups of motions, homotheties, conformals and affines, and to prove a theorem on the relationship between the first three of these. The basic theories and notations of differential geometry are set up first, and a useful theorem on first-order partial differential equations is proved. The concepts of General Relativity are introduced, space-times are defined and a brief account of the well-known Petrov and Segre classifications is given. The interplay between these classifications and the isotropy structure of the various Lie groups is discussed as is the so-called 'Schmidt method'. Generalised p.p. waves are studied, with a special study of the subclass of generalised plane waves undertaken, many different characterisations of these latter are found and their admitted symmetries are completely described. Motions, homotheties and affines are considered. A survey of symmetries in Minkowski space, and a summary of known results on space-times with high dimensional groups of motions is given. The problem of r-dimensional groups of homotheties is studied. The r 6 cases are completely resolved, and examples in the r = 5 cases are given. All examples of non-flat space-times admitting the maximal group of affines are displayed, correcting an error in the literature. The thesis ends with a proof of the Bilyalov-Defrise-Carter theorem, which states that for any non conformally flat space-time there is a conformally related metric for which the original group of conformals is a group of homotheties (motions if not conformal to generalised plane waves). The proof given does not use Bilyalov's analyticity assumption, and is more geometric than Defrise-Carter. The maximum size of the conformal group for a given Petrov type is found. An appendix gives a brief account of some REDUCE routines used to check some algebraic manipulations.

by Tsoi, Man. / Thesis submitted in: July 1999. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 56-57). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Preliminary --- p.12 / Chapter 2.1 --- Lie Group and Lie Algebra --- p.12 / Chapter 2.2 --- Harmonic Maps --- p.15 / Chapter 2.3 --- Some Factorization theorems --- p.17 / Chapter 3 --- A Survey on Unlenbeck's Results --- p.22 / Chapter 3.1 --- Preliminary --- p.24 / Chapter 3.2 --- Extended Solutions --- p.26 / Chapter 3.3 --- The Variational Formulas for the Extended Solutions --- p.30 / Chapter 3.4 --- "The Representation of A(S2, G) on holomorphic maps C* → G" --- p.33 / Chapter 3.5 --- An Action of G) on extended solutions and Backlund Transformations --- p.39 / Chapter 3.6 --- The Additional S1 Action --- p.42 / Chapter 3.7 --- Harmonic Maps into Grassmannians --- p.43 / Chapter 4 --- Harmonic Maps into Compact Lie Groups --- p.47 / Chapter 4.1 --- Symmetry group of the harmonic map equation --- p.48 / Chapter 4.2 --- A New Formulation --- p.49 / Chapter 4.3 --- "Harmonic Maps into Grassmannian, Another Point of View" --- p.53 / Bibliography