## Conservation laws and exact solutions for some nonlinear partial differential equations

##### Abstract

In this thesis we study some nonlinear partial differential equations which appear
in several physical phenomena of the real world. Exact solutions and conservation
laws are obtained for such equations using various methods. The equations
which are studied in this work are a generalized coupled (2+1)-dimensional hyperbolic
system, a modified Kortweg-de Vries type equation, the higher-order modi
fied Boussinesq equation with damping term, coupled Korteweg-de Vries equations,
coupled Boussinesq equations, a generalized Zakharov-Kuznetsov equation,
a generalized Ablowitz-Kaup-Newell-Segur equation and a potential Kadomtsev-
Petviashvili equation with p-power nonlinearity.
We perform a complete Lie symmetry classifcation of a generalized coupled (2+1)-
dimensional hyperbolic system, which models many physical phenomena in nonlinear
sciences. The Lie group classifcation of the system provides us with elevendimensional
equivalence Lie algebra and has several possible extensions. It is further
shown that several cases arise in classifying the arbitrary parameters, the
forms of which include amongst others the power and exponential functions.
We obtain exact solutions of two nonlinear evolution equations, namely, modified
Kortweg-de Vries equation and higher-order modified Boussinesq equation with
damping term. The (G0=G)-expansion method is employed to obtain the exact
solutions. Travelling wave solutions of three types are obtained and these are the
solitary waves, periodic and rational. In addition, the conservation laws for higherorder
modified Boussinesq equation with a damping term are constructed using the
multiplier approach.
The (G0=G)-expansion method is employed to derive the exact travelling wave
solutions of coupled Korteweg-de Vries equations. The solutions obtained include
the soliton solutions. Furthermore, the conservation laws for these equations are
obtained.
Travelling wave solutions of coupled Boussinesq equations are determined and conservation
laws are obtained for the system using the new conservation theorem and
multiplier approach.
We study a generalized Zakharov-Kuznetsov equation in three variables, which has
applications in the nonlinear development of ion-acoustic waves in a magnetized
plasma. Conservation laws for this equation are constructed using the new conservation
theorem. Furthermore, new exact solutions are obtained by employing the
Lie symmetry method along with the simplest equation method.
Conservation laws of a generalized Ablowitz-Kaup-Newell-Segur equation are constructed
by using Noether theorem. The exact solutions are obtained using the Lie
symmetry method together with the simplest equation method and direct integration.
Finally, a potential Kadomtsev-Petviashvili equation with p-power nonlinearity,
which arises in a number of significant nonlinear problems of physics and applied
mathematics is studied. We carry out Noether symmetry classifcation on this
equation. Four cases arise depending on the values of p and consequently we
construct conservation laws for these cases with respect to the second-order Lagrangian.
In addition, exact solutions for this equation are obtained using the Lie
group analysis together with the Kudryashov method and direct integration.