Integrability and Solutions of the (2 + 1)-dimensional Hunter–Saxton Equation*

In this paper, the (2 + 1)-dimensional Hunter-Saxton equation is proposed and studied. It is shown that the (2 + 1)-dimensional Hunter–Saxton equation can be transformed to the Calogero–Bogoyavlenskii–Schiff equation by reciprocal transformations. Based on the Lax-pair of the Calogero–Bogoyavlenskii–Schiff equation, a non-isospectral Lax-pair of the (2 + 1)-dimensional Hunter–Saxton equation is derived. In addition, exact singular solutions with a finite number of corners are obtained. Furthermore, the (2 + 1)-dimensional μ-Hunter–Saxton equation is presented, and its exact peaked traveling wave solutions are derived.