On the Eckhaus and the Benjamin-Feir Instabilty in the Vicinity of a Tricritical Point

For pattern forming systems with spatial variations in one direction the Eckhaus instability, the instability against spatial modulations of the pattern, has been studied for a stationary forward bifurcation for a number of systems theoretically1–5 and experimentally6,7 and a characteristic band of wavelengths, which are linearly stable against spatial modulations, emerges in all these cases. Here we perform the analogous analysis at the tricritical point, at which the coefficient of the cubic term in the envelope equation vanishes and where we assume saturation to quintic order. It is found that the band of Eckhaus stable wavelengths is a factor of \( {{(1.5)}^{{\tfrac{1}{2}}}} \) wider in this case. We also summarize the results obtained for the Eckhaus instability recently in the case of a weakly inverted stationary bifurcation8, where the envelope equation takes the form $$ \dot{A} = \in A + \gamma {{A}_{{xx}}} + \beta |A{{|}^{2}}A - \delta |A{{|}^{4}}A $$ (1) and where ϵ, β, γ, and δ are real with β > 0 for a weakly inverted bifurcation.