Computing the Wave Speed of Soliton-Like Solutions in SmC* Liquid Crystals

Abstract

We present a novel method for numerically computing the wave speed of a soliton-like travelling wave in chiral smectic C liquid crystals (SmC*) that satisfies a parabolic partial differential equation (PDE) with a general nonlinear term [1]. By transforming the PDE to a co-moving frame and recasting the resulting problem in phase-space, the original PDE can be expressed as an integral equation known as an exceptional nonlinear Volterra-type equation of the second kind. This technique is motivated by, but distinct in nature from, iterative integral methods introduced by Chernyak [2]. By applying a simple trapezoidal method to the integral equation we generate a system of nonlinear simultaneous equations which we solve for our phase plane variable at equally spaced intervals using Newton iterates. The equally spaced phase variable solutions are then used to compute the wave speed of the associated travelling wave. We demonstrate an algorithm for performing the necessary calculations by considering an example from liquid crystal theory, where a parabolic PDE with a nonlinear reaction term has a solution and wave speed which are known exactly [3,4]. The analytically derived wave speed is then compared with the numerically computed wave speed using our new scheme.