2009/01/01 by Giuseppe Alı̀, ,Dipartimento di Matematica, Universitá della Calabria and INFN-Gruppo c. Cosenza, I-87036 Arcavacata di Rende (CS), John K. Hunter · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Classical mechanics #Field (mathematics) #Geometry #Inertia #Integrable system #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Orientation (vector space) #Physics #Quadratic growth #Quantum mechanics #Twist
paper · pdf · doi:10.3934/krm.2009.2.1
openalex publication_date 2009/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/21
We study a variational system of nonlinear hyperbolic partial differential equationsthat describes the propagation of orientation waves in a director fieldwith rotational inertia and potential energy given by the Oseen-Frankenergy 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 describedby the quadratically nonlinear Hunter-Saxton equation.In this paper, we derive a new cubically nonlinearasymptotic equation that describesweakly nonlinear twist waves. This equation providesa surprising representation of the Hunter-Saxton equation, andlike the Hunter-Saxton equation it is completely integrable.There are analogous cubically nonlinear representations of the Camassa-Holmand Degasperis-Procesi equations.Moreover, two different, but compatible, variational principles for the Hunter-Saxtonequation arise from a single variational principlefor the primitive director field equations in the two differentlimits for splay and twist waves. We also usethe asymptotic equation to analyzea one-dimensional initial value problem for thedirector-field equations with twist-wave initial data.