The stability of travelling waves induced by crossed electric and magnetic fields in nematic liquid crystals

A theoretical study is carried out into the stability of travelling wave solutions to an approximate dynamic equation for the problem in which a nematic liquid crystal is subjected to crossed electric and magnetic fields. The authors recently found three types of travelling wave solutions for this problem [2], each characterised by the control parameter $q$ which describes the relationship between the magnitudes of the fields and their crossed angle. Two types of stability are ex amined: the first considers perturbations which vanish outside some finite interval in the moving coordinate of the travelling wave, while the second considers quite general perturbations belonging to a weighted $L_2({\bf R})$ space, the weighting function being determined by the particular solution and the control parameter $q$. When the first type of stability occurs, perturbations decay to zero as time increases. In the second type of stability perturbations may eith er decay to zero or induce a small phase shift to the original travelling wave. Both these versions of stability depend crucially on $q$ and on the type of travelling wave solution being considered.