In this paper, we develop, analyze and test a local discontinuous Galerkin (LDG) method for solving the Hunter-Saxton (HS) equation which contains nonlinear high order derivatives and its zero-viscosity and zero-dispersion limit. The energy stability for general solutions are proved and numerical simulation results for different types of solutions of the nonlinear HS equation are provided to il...

In this paper, we present infinitely many conserved densities satisfying particular conservation law $F_{t}=(2uF)_{x}$ for the generalized Riemann equations at $N=2,3,4$. $N=2$ case, also construct corresponding to new laws containing an arbitrary smooth function. virtue of reductions and/or changes variables, related are obtained two component Hunter-Saxton equation, Gurevich-Zybin equation an...

We consider N=2 supersymmetric extensions of the Camassa-Holm and HunterSaxton equations. We show that they admit geometric interpretations as Euler equations on the superconformal algebra of contact vector fields on the 1|2dimensional supercircle. We use the bi-Hamiltonian formulation to derive Lax pairs. Moreover, we present some simple examples of explicit solutions. As a byproduct of our an...