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.

Orientation waves in a director field with rotational inertia

ALI', Giuseppe;
2009

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.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11770/154138
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? 25
social impact