Mathematics Colloquia and Seminars

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We will consider orientation waves in a massive director field of unit vectors. The director field satisfies a variational system of wave equations in which the kinetic energy arises from the rotation of the director and the potential energy is given by the Oseen-Frank energy for liquid crystals. There are two types of orientation waves: genuinely nonlinear "splay" waves, described asymptotically by the Hunter-Saxton equation; and linearly degenerate "twist waves", described asymptotically by a cubically nonlinear form of the Hunter-Saxton equation. We will show that twist waves generate splay waves, and construct an asymptotic solution of the initial-value problem in one space dimension with inital data consisting of a compactly supported, small-amplitude twist wave. The system of equations for massive director fields belongs to a class of variational hyperbolic systems of wave equations which includes the vacuum Einstein equations of general relativity as a highly degenerate member. If time permits, we will contrast the properties of the director field equations with those of the Einstein equations.