Kinetic and Related Models (KRM)

Orientation waves in a director field with rotational inertia

Pages: 1 - 37, Volume 2, Issue 1, March 2009      doi:10.3934/krm.2009.2.1

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Giuseppe Alì - Dipartimento di Matematica, Universitá della Calabria and INFN-Gruppo c. Cosenza, I-87036 Arcavacata di Rende (CS), Italy (email)
John K. Hunter - Department of Mathematics, University of California at Davis, Davis CA 95616, United States (email)

Abstract: We study a variational system of nonlinear hyperbolic partial differential equations that describes the propagation of orientation waves in a director field with rotational inertia and potential energy given by the Oseen-Frank energy from the continuum theory of nematic liquid crystals. There are two types of waves, which we call splay and twist waves, respectively. Weakly nonlinear splay waves are described by the quadratically nonlinear Hunter-Saxton equation. In this paper, we derive a new cubically nonlinear asymptotic equation that describes weakly nonlinear twist waves. This equation provides a surprising representation of the Hunter-Saxton equation, and like the Hunter-Saxton equation it is completely integrable. There are analogous cubically nonlinear representations of the Camassa-Holm and Degasperis-Procesi equations. Moreover, two different, but compatible, variational principles for the Hunter-Saxton equation arise from a single variational principle for the primitive director field equations in the two different limits for splay and twist waves. We also use the asymptotic equation to analyze a one-dimensional initial value problem for the director-field equations with twist-wave initial data.

Keywords:  Nonlinear hyperbolic waves. Liquid crystals. Variational principles. Integrable Hamiltonian PDEs.
Mathematics Subject Classification:  Primary: 35L70, 37K10, 74J30.

Received: November 2008;      Revised: November 2008;      Available Online: January 2009.