On periodically modulated rolls in the generalized Swift–Hohenberg equation: Galerkin’ approximations

Abstract

We study in this paper the existence of periodically modulated in one variable and localized in another variable solutions to the cubic Swift–Hohenberg equation on the plane R2. In the first part we try to apply the method by Kirschgässner–Mielke to reduce the problem to the search of finite dimensional submanifolds with periodic orbits on them in some formal infinite-dimensional dynamical system generated by the stationary SH equation. It turns out that it cannot be done immediately due to properties of the spectrum for the linearized system at the localized roll (a one-dimensional pulse). In the second part we change roles of variables and formulate the problem as funding homoclinic orbits to an equilibrium of the formal infinite-dimensional system in the space of periodic functions in variable y. Staying apart the proof of the exact theorem on the existence of related center manifold, we exploit the Bubnov–Galerkin method to derive the Hamiltonian finite-dimensional ODEs with four or six degrees of freedom having the saddle type equilibrium whose homoclinic orbits correspond to approximate solution on needed type. The search for homoclinic orbits is performed by means of numerical methods.