Multiple layered solutions of the nonlocal bistable equation

The nonlocal bistable equation is a model proposed recently to study materi als whose constitutive relations among the variables that describe their st ates are nonlocal. It resembles the local bistable equation (the Allen-Cahn equation) in some way, but contains a much richer set of solutions. In thi s paper we consider two types of solutions. The first are the periodic solu tions on a finite interval. These solutions are observed in materials like elastic crystals undergoing martensitic phase transitions and diblock copol ymers at low temperatures. They are constructed by a variational method kno wn as the Gamma -limit technique. The second are solutions on the entire re al line with transition layers, which are found by the formal matched asymp totics argument. We construct them to compare with the single layer heteroc linic and traveling wave solutions of the local bistable equation. The exis tence of multiple layered solutions depends on a unique nonlocal feature: t he presence of two properly balanced competing effects of the constitutive relation, the oscillation inhibiting effect and the oscillation forcing eff ect, which coexist at two different length scales. (C) 2000 Elsevier Scienc e B.V. All rights reserved.