Generalized homoclinic solutions for the Swift–Hohenberg equation

This paper considers the Swift–Hohenberg equation u ⁗ + β u ″ + u 3 − u = 0 , β > 0 . Applying a perturbation method and adjusting some phase shift constants, we prove that it has a homoclinic solution connecting a periodic solution (called generalized homoclinic solution, thereafter) near the origin for each positive constant β, which is a new result.