Some Canonical Bifurcations in the Swift-Hohenberg Equation

Exact solutions of the cubic–quintic Swift–Hohenberg equation and their bifurcations

Continuation and Bifurcation of Grain Boundaries in the Swift-Hohenberg Equation

We study grain boundaries between striped phases in the prototypical Swift-Hohenberg equation. We propose an analytical and numerical far-field-core decomposition that allows us to study existence and bifurcations of grain boundaries analytically and numerically using continuation techniques. This decomposition overcomes problems with computing grain boundaries in a large doubly periodic box wi...

Attractor Bifurcation and Final Patterns of the N-dimensional and Generalized Swift-hohenberg Equations

In this paper I will investigate the bifurcation and asymptotic behavior of solutions of the Swift-Hohenberg equation and the generalized Swift-Hohenberg equation with the Dirichlet boundary condition on a onedimensional domain (0, L). I will also study the bifurcation and stability of patterns in the n-dimensional Swift-Hohenberg equation with the odd-periodic and periodic boundary conditions....

Rigorous Verification of Bifurcations of Differential Equations via the Conley Index Theory

We propose a new approach for capturing bifurcations of (semi)flows by using a topological tool, the Conley index. We can apply this concept to capture bifurcations with rigorous numerics. As an example, we consider the dynamics generated by the Swift–Hohenberg PDE and show that a pitchfork-like bifurcation occurs in a certain region.

A ug 2 01 1 Localised states in an extended Swift – Hohenberg equation

Recent work on the behaviour of localised states in pattern forming partial differential equations has focused on the traditional model Swift-Hohenberg equation which, as a result of its simplicity, has additional structure — it is variational in time and conservative in space. In this paper we investigate an extended Swift-Hohenberg equation in which non-variational and non-conservative effect...