The stability of travelling waves induced by crossed electric and magnetic fields in nematic liquid crystals
This paper considers the time-dependent solution to the dynamic equation for the director \({\mathbf n}\) in a sample of nematic liquid crystal, which is infinite in the \(z\)-direction, subject to crossed electric \({\mathbf E}\) and magnetic \({\mathbf H}\) fields applied in the \(xy\)-plane when bulk flow is neglected. Seeking solutions of the form \({ \mathbf n}= (\cos \varphi (z,t),\sin \varphi (z,t),0)\) with \({\mathbf E}= E(\cos \beta, \sin \beta, 0)\) and \({\mathbf H}= H(1,0,0)\), where \(E\) and \(H\) are constant field strengths, the equation for \({\mathbf n}\) reduces to \(\gamma_1 \varphi_t= K_2 \varphi_{zz} -{1\over 2} \chi_a H^2 \sin 2\varphi- {1\over 2} \varepsilon_a \varepsilon_0 E^2 \sin 2(\varphi- \beta)\), where \(\gamma_1\) is the twist viscosity coefficient, \(K_2\) is the elastic twist constant, \(\chi_a\) is the diamagnetic anisotropy, \(\varepsilon_a\) is the dielectric anisotropy and \(\varepsilon_0\) is the permittivity of free space. After a suitable rescaling and a suitable cubic approximation to \(\varphi\) for the sine terms in the above equation, the methods of Lam [\textit{I. W. Stewart} and \textit{T. R. Faulkner}, J. Phys. A, Math. Gen. 28, No. 19, 5643-5652 (1995)] are applied to obtain three distinct types of travelling waves of the resulting equation. These solutions are characterised by a so-called control parameter \(q\), which links \(E,H\) and \(\beta\), since its magnitude controls the amplitudes of these travelling waves. The authors proceed to examine two types of stability for each type of travelling wave. One considers perturbations that vanish outside some finite interval in the moving coordinate of the travelling wave, and the other considers general perturbations belonging to a weighted \(L_2 ({\mathbf R})\) space, the weighting function being determined by the type of travelling wave and \(q\). It is found that both types of stability depend crucially on \(q\) and the type of travelling wave being considered.
scientific article; zbMATH DE number 1081103