2006/09/07 by Giuseppe Alı̀, Giuseppe Ali, John K. Hunter +2 · 1 citation
Mathematics · Physics and Astronomy · #35L70 #37K10 #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #math-ph #math.AP #math.MP #msc:35L70 #msc:37K10
paper · pdf · doi:10.48550/arxiv.math/0609189
arxiv created 2006/09/07 · openalex publication_date 2006/09/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the propagation of orientation waves in a director field with rotational inertia and potential energy given by the Oseen-Frank energy functional from the continuum theory of nematic liquid crystals. There are two types of waves, which we call splay and twist waves. Weakly nonlinear splay waves are described by the quadratically nonlinear Hunter-Saxton equation. Here, we show that weakly nonlinear twist waves are described by a new cubically nonlinear, completely integrable asymptotic equation. This equation provides a surprising representation of the Hunter-Saxton equation as an advection equation. There is an analogous representation of the Camassa-Holm equation. We use the asymptotic equation to analyze a one-dimensional initial value problem for the director-field equations with twist-wave initial data.