Orientation waves in a director field with rotational inertia

Abstract

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 for u as an advection equation for v, where u xx = v 2x . 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.Proof. Writing (1.4) in terms of characteristic coordinates (ξ, τ ) in which τ = t and x τ = u, we find that the PDE becomeswhere F , G are functions of integration, andThe elimination of U from these equations yields a PDE for J,

Generic regularity and Lipschitz metric for the Hunter–Saxton type equations

“…Here the variable u = u(t, x) describes the horizontal velocity of the fluid, ρ = ρ(t, x) describes the horizontal deviation of the surface from equilibrium. When ρ ≡ 0, the above system becomes the one-component Hunter-Saxton equation 2) which is an asymptotic equation of the variational wave equation used to model nematic liquid crystal [1,11]. The Hunter-Saxton equation (1.2) was first derived in [11] as an asymptotic equation of the variational wave equation, which was considered in [1,2,3,4,8,21,22], for the nematic liquid crystals.…”

“…When ρ ≡ 0, the above system becomes the one-component Hunter-Saxton equation 2) which is an asymptotic equation of the variational wave equation used to model nematic liquid crystal [1,11]. The Hunter-Saxton equation (1.2) was first derived in [11] as an asymptotic equation of the variational wave equation, which was considered in [1,2,3,4,8,21,22], for the nematic liquid crystals. The global existences of weak conservative and dissipative solutions of (1.2) were first proved by Hunter and Zheng in [12,13] on the initial value problem, by studying the self-similar solutions, then were treated by several other methods including the Young measure method by Zhang and Zheng in [20], and the characteristic method by Bressan and Constantin [5] and Bressan, Zhang and Zheng [7] on the initial value or initial boundary value problem.…”

Singularity and existence for a wave system of nematic liquid crystals

“…The mean orientation of the long molecules in a nematic liquid crystal is described by a director field of unit vectors, n ∈ S 2 , and the propagation of the orientation waves in the director field could be modelled by below Euler-Largrangian equations derived from the least action principle [12,1], (1.1) n tt + µn t + δW (n, ∇n) δn = λn, n · n = 1, where the well-known Oseen-Franck potential energy density W is given by (1.2) W (n, ∇n) = 1 2 α(∇ · n) 2 + 1 2 β (n · ∇ × n) 2 + 1 2 γ |n × (∇ × n)| 2 . The positive constants α, β, and γ are elastic constants of the liquid crystal, corresponding to splay, twist, and bend, respectively.…”

Nonlinear variational surface waves

“…One example of a variational system of the form (1) arises as a description of orientation waves in a massive director field [2,19]. The orientation of the director is described by a unit vector field…”

“…The director-field system has two types of bulk waves, called splay and twist waves [2], which are analogous to longitudinal p-waves and transverse s-waves, respectively, in elasticity. It is interesting to compare the nonlinear behavior of longitudinal, transverse, and surface waves in these two systems.…”