Symmetry analysis, conservation laws and exact solutions of certain nonlinear partial differential equations
Abstract
In this research work we study some nonlinear partial differential equations which
model many physical phenomena in science, engineering and finance. Closedform
solutions and conservation laws are obtained for such equations using various
methods. The nonlinear partial differential equations that are investigated
in this thesis are; a variable coeffcients Gardner equation, a generalized (2+1)-
dimensional Kortweg-de Vries equation, a coupled Korteweg-de Vries-Burgers system,
a Kortweg-de Vries{modified Kortweg-de Vries equation, a generalized improved
Boussinesq equation, a Kaup-Boussinesq system, a classical model of Prandtl's
boundary layer theory for radial viscous
ow, a generalized coupled (2+1)-dimensional
Burgers system, an optimal investment-consumption problem under the constant
elasticity of variance model and the Zoomeron equation.
We perform Lie group classi cation of a variable coe cients Gardner equation,
which describes various interesting physics phenomena, such as the internal waves
in a strati ed ocean, the long wave propagation in an inhomogeneous two-layer
shallow liquid and ion acoustic waves in plasma with a negative ion. The Lie
group classi cation of the equation provides us with four-dimensional equivalence
Lie algebra and has several possible extensions. It is further shown that several
cases arise in classifying the arbitrary parameters. Conservation laws are obtained
for certain cases.
A generalized (2+1)-dimensional Korteweg-de Vries equation is investigated. This
equation was recently constructed using Lax pair generating technique. The extended
Jacobi elliptic method is employed to construct new exact solutions for this
equation and obtain cnoidal and snoidal wave solutions. Moreover, conservation
laws are derived using the multiplier method.