Lie symmetry analysis, optimal system, exact solutions and conservation laws of a class of high-order nonlinear wave equations

The symmetries, conservation laws and exact solutions to the nonlinear partial differential equations play a significant role in nonlinear science and mathematical physics. Symmetry is derived from physics, and it is a mathematical description for invariance. Symmetry group theory plays an important role in constructing explicit solutions, whether the equations are integrable or not. By using the symmetry method, an original nonlinear system can be reduced to a system with fewer independent variables through any given subgroup. But, since there are almost always an infinite number of such subgroups, it is usually not feasible to list all possible group invariant solutions to the system. It is anticipated to find all those equivalent group invariant solutions, that is to say, to construct the one-dimensional optimal system for the Lie algebra. Construction of explicit forms of conservation laws is meaningful, as they are used for developing the appropriate numerical methods and for making mathematical analyses, in particular, of existence, uniqueness and stability. In addition, the existence of a large number of conservation laws of a partial differential equation (system) is a strong indication of its integrability. The similarity solutions are of importance for investigating the long-time behavior, blow-up profile and asymptotic phenomena of a non-linear system. For instance, in some circumstance, the asymptotic behaviors of finite-mass solutions of non-linear diffusion equation with non-linear source term are described by an explicit self-similar solution, etc. However, how to tackle these matters is a complicated problem that challenges researchers to be solved. In this paper, by using the symmetry method, we obtain the symmetry reduction, optimal systems, and many new exact group invariant solution of a fifth-order nonlinear wave equation. By Lie symmetry analysis method, the point symmetries and an optimal system of the equation are obtained. The exact power series solutions to the equation are provided by the power series method, such solutions can be used for numerical computations in both theory and physical applications conveniently. Finally, a lot of conservation laws of the fifth-order nonlinear wave equation are presented by using the adjoint equation and symmetries of the equation.