On the symmetry analysis of some wave-type nonlinear partial differential equations

Abstract

In this work, we study the applications of Lie symmetry analysis to certain nonlinear wave equations. Exact solutions and conservation laws are obtained for such equations. The equations which are considered in this thesis are the generalized ( 2+ 1 )dimensional Klein-Gordon equation, the generalized double sinh-Gordon equation, the generalized double combined sinh-cosh-Gordon equation, the (2+ 1 )-dimensional nonlinear sinh-Gordon equation, the (3+ 1 )-dimensional nonlinear sinh-Gordon equat ion, the Boussinesq-double sine-Gordon equation, the Boussinesq-double sinh-Gordon equation, the Boussinesq-Liouville type I equation and the Boussinesq-Liouville type II equation. The generalized (2+ 1 )-dimensional Klein-Gordon equation is investigated from the point of view of Lie group classification. We show that this equation admits a nine-dimensional equivalence Lie algebra. It is also shown that the principal Lie algebra consists of six symmetries. Several possible extensions of the principal Lie algebra are computed and the group-invariant solutions-of the generalized (2+ 1 )-dimensional Klein-Gordon equation are presented for power law and exponential function cases. Thereafter, we illustrate that the generalized (2+ 1 )-dimensional Klein-Gordon equation is non-linearly self-adjoint. In addition, we derive conservation laws for the non-linearly self-adjoint sub-classes by using the new Ibragimov theorem. Lie symmetry method is performed on a generalized double sinh-Gordon equation. Exact solutions of a generalized double sinh-Gordon equation are obtained by using the Lie symmetry method in conjunction with the simplest equation method and the exponential function method. In addition to exact solutions we also present conservation laws which are derived using four different methods, namely the direct method, the Noether theorem, the new conservation theorem due to Ibragimov and the multiplier method. The generalized double combined sinh-cosh-Gordon equation is investigated using Lie group analysis. Exact solutions are obtained using the Lie group method together with the simplest equation method. Conservation laws are also obtained by using four different approaches, namely the direct method, the Noether theorem, the new conservation theorem due to Ibragimov and the multiplier method for the underlying equation. The (2+ 1 )-dimensional nonlinear sinh-Gordon equation and the (3+ 1 )-dimensional nonlinear sinh-Gordon equation are investigated by using Lie symmetry analysis. The similarity reductions and exact solutions with the aid of simplest equation method and ( G' / G)-expansion methods are computed. In addition to exact solutions, the conservation laws are derived as well for both the equations. Finally, the four Boussinesq-type equations, namely, the Boussinesq-double sineGordon equation, t he Boussinesq-double sinh-Gordon equation, the Boussinesq-Liouville type I equation and the Boussinesq-Liouville type II equation are analysed using Lie group analysis. Exact solutions for these equations are obtained using the Lie symmetry method in conjunction with the simplest equation. Conservation laws are also obtained for these equations by employing two methods, namely, the Noether theorem and the multiplier method.