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

Physica D: Nonlinear Phenomena. 2023. Vol. 454. Article 133845.

Kulagin N. E., L.M. Lerman

We study in this pap er 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 Kirschgassner-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 of needed type. The search for homoclinic orbits is performed by means of numerical methods.
 

Keywords: approximation, periodical, Localized solutions

Publication based on the results of: Теория динамических систем и ее приложения (2023)