Several Types of Similarity Solutions for the Hunter–Saxton Equation*

Abstract

Abstract We study separable and self-similar solutions to the Hunter–Saxton equation, a nonlinear wave equation which has been used to describe an instability in the director Geld of a nematic liquid crystal (among other applications). Essentially, we study solutions which arise from a nonlinear inhomogeneous ordinary differential equation which is obtained by an exact similarity transform for the Hunter–Saxton equation. For each type of solution, we are able to obtain some simple exact solutions in closed-form, and more complicated solutions through an analytical approach. We find that there is a whole family of self-similar solutions, each of which depends on an arbitrary parameter. This parameter essentially controls the manner of self-similarity and can be chosen so that the self-similar solutions agree with given initial data. The simpler solutions found constitute exact solutions to a nonlinear partial differential equation, and hence are also useful in a mathematical sense. Analytical solutions demonstrate the variety of behaviors possible within the wider family of similarity solutions. Both types of solutions cast light on self-similar phenomenon arising in the Hunter–Saxton equation.