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
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:nwu/oai:dspace.nwu.ac.za:10394/16146 |
| Date | January 2013 |
| Creators | Adem, Abdullahi Rashid |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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