The soliton-stripe pattern in the Seul–Andelman membrane

The Seul–Andelman membrane is a system of two coupled fields: the composition φ of one of the two (A and B) constitutive molecules, and the height profile h of the flexible membrane. The free energy of the system consists of two parts. The first part is the usual Ginzburg–Landau free energy of φ; the second part is attributed to the bending of the membrane and the coupling of φ to h. The coupling term models the tendency that the two molecular constituents display an affinity for regions of the membrane of different local curvature. In a particular parameter range we prove the existence of the soliton-stripe pattern, using the Γ-limit theory in perturbative variational calculus. This pattern, modeled by one-dimensional local minimizers of the free energy of the system, consists of A-rich and B-rich stripes covering the membrane, delineated by sharp domain walls. The optimal spacing between domain walls is determined from the global minimizer of the Γ-limit.