Symmetry classification and conservation laws 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 hyperbolic Lane-Emden system, a generalized hyperbolic Lane-Emden system, a coupled Jaulent-Miodek system and a (2+1)-dimensional Jaulent-Miodek equation power-law nonlinearity. We carry out a complete Noether and Lie group classification of the radial form of a coupled system of hyperbolic equations. From the Noether symmetries we establish the corresponding conserved vectors. We also determine constraints that the non-linearities should satisfy in order for the scaling symmetries to be Noetherian. This led us to a critical hyperbola for the systems under consideration. An explicit solution is also obtained for a particular choice of the parameters. We perform a complete Noether symmetry analysis of a generalized hyperbolic LaneEmden system. Several constraints for which Noether symmetries exist are derived. In addition, we construct conservation laws associated with the admitted Noether symmetries. Thereafter, we briefly discuss the physical meaning of the derived conserved vectors. We carry out a complete group classification of a generalized coupled hyperbolic Lane-Emden system. It is shown that the underling system admits six-dimensional equivalence Lie algebra. We further show that the principle Lie algebra which is one dimensional extends in several cases. We also carry out Lie reductions for some cases. Symmetry analysis is performed on a coupled Jaulent-Miodek system, which arises in many branches of physics such as particle physics and fluid dynamics. The similarity reductions and new exact solutions are constructed. Subsequently, con-servation laws are derived using the multiplier approach. We study complete Noether symmetry classification of a (2+1)-dimensional Jaulent-Miodek equation with power-law nonlinearity. Conservation laws for several cases which admit Noether point symmetries are established.