Transverse stability of the one-dimensional kink solution of the discrete Cahn-Hilliard equation

We give an analysis of a discrete version of the Cahn-Hilliard equation, which admits a one-dimensional kink solution. The stability of such a kink solution to perpendicular perturbations is analyzed using an asymptotic matching method as used by Bettinson and Rowlands [Phys. Rev. E 54, 6102 (1996)] and a Green's function technique similar to that used by Shinozaki and Oono [Phys. Rev. E 47, 804 (1993)]. The kink is found to be stable in both cases, but we find that the two methods do not agree, at first order in the growth rate, for perturbations of wave number close to k=2p pi (where p is some integer). Reasons for this disagreement and why the method given by Bettinson and Rowlands leads to the correct result are given. We then use equations derived in by Bettinson and Rowlands to compare stability results to those obtained for the continuous version of the equation. We also analyze the stability to perturbations of wave number close to k=(2p + 1)pi and finally, using a Pade approximant, we give an expression for the growth rate of perturbations of all wavelengths. Our results quantify the difference between the continuum and discrete cases.