American Institute of Mathematical Sciences

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March  2009, 2(1): 1-37. doi: 10.3934/krm.2009.2.1

Orientation waves in a director field with rotational inertia

Giuseppe Alì 1, and  John K. Hunter 2,

1. 

Dipartimento di Matematica, Universitá della Calabria and INFN-Gruppo c. Cosenza, I-87036 Arcavacata di Rende (CS), Italy
2. 

Department of Mathematics, University of California at Davis, Davis CA 95616, United States

Received  November 2008 Revised  November 2008 Published  January 2009

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, 74J3.
Citation: Giuseppe Alì, John K. Hunter. Orientation waves in a director field with rotational inertia. Kinetic & Related Models, 2009, 2 (1) : 1-37. doi: 10.3934/krm.2009.2.1
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