On the symmetry analysis of some wavetype 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 KleinGordon equation, the generalized double sinhGordon equation, the generalized double combined sinhcoshGordon equation, the (2+ 1 )dimensional nonlinear sinhGordon equation, the (3+ 1 )dimensional nonlinear sinhGordon equat ion, the Boussinesqdouble sineGordon equation, the Boussinesqdouble sinhGordon equation, the BoussinesqLiouville type I equation and the BoussinesqLiouville type II equation. The generalized (2+ 1 )dimensional KleinGordon equation is investigated from the point of view of Lie group classification. We show that this equation admits a ninedimensional 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 groupinvariant solutionsof the generalized (2+ 1 )dimensional KleinGordon equation are presented for power law and exponential function cases. Thereafter, we illustrate that the generalized (2+ 1 )dimensional KleinGordon equation is nonlinearly selfadjoint. In addition, we derive conservation laws for the nonlinearly selfadjoint subclasses by using the new Ibragimov theorem. Lie symmetry method is performed on a generalized double sinhGordon equation.
Exact solutions of a generalized double sinhGordon 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 sinhcoshGordon 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 sinhGordon equation and the (3+ 1 )dimensional nonlinear sinhGordon 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 Boussinesqtype equations, namely, the Boussinesqdouble sineGordon equation, t he Boussinesqdouble sinhGordon equation, the BoussinesqLiouville type I equation and the BoussinesqLiouville 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.
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