Symmetry Analysis and Conservation Laws for the Hunter-Saxton Equation

In this paper, the problem of determining the most general Lie point symmetries group and conservation laws of a well known nonlinear hyperbolic PDE in mathematical physics called the Hunter-Saxton equation (HSE) is analyzed. By applying the basic Lie symmetry method for the HSE, the classical Lie point symmetry operators are obtained. Also, the algebraic structure of the Lie algebra of symmetries is discussed and an optimal system of one-dimensional subalgebras of the HSE symmetry algebra which creates the preliminary classification of group invariant solutions is constructed. Particularly, the Lie invariants as well as similarity reduced equations corresponding to infinitesimal symmetries are obtained. Mainly, the conservation laws of the HSE are computed via three different methods including Boyer's generalization of Noether's theorem, first homotopy method and second homotopy method.