Wriggled Lamellar Solutions and Their Stability in the Diblock Copolymer Problem

In a diblock copolymer system the free energy field depends non-locally on the monomerdensity field. In addition there are two positive parameters in the constitutive relation. One ofthem is small with respect to which we do singular perturbation analysis. The second one isof order 1 with respect to which we do bifurcation analysis. Combining the two techniques wefind wriggled lamellar solutions of the Euler-Lagrange equation of the total free energy undera hypothesis regarding the simplicity of the principal eigenvalue, which is generically satisfied.The wriggled lamellar solutions bifurcate from the perfect lamellar solutions. The stability ofthe wriggled lamellar solutions is reduced to a relatively simple finite dimensional problem,which may be solved accurately by a numerical method. Our tests show that most of themare stable. The existence of such stable wriggled lamellar solutions explains why in reality thelamellar phase is fragile and it often exists in distorted forms.