  We study the propagation of orientation waves in a director field with
rotational inertia and potential energy given by the Oseen-Frank energy
functional from the continuum theory of nematic liquid crystals. There are two
types of waves, which we call splay and twist waves. Weakly nonlinear splay
waves are described by the quadratically nonlinear Hunter-Saxton equation.
Here, we show that weakly nonlinear twist waves are described by a new
cubically nonlinear, completely integrable asymptotic equation. This equation
provides a surprising representation of the Hunter-Saxton equation as an
advection equation. There is an analogous representation of the Camassa-Holm
equation. We use the asymptotic equation to analyze a one-dimensional initial
value problem for the director-field equations with twist-wave initial data.
