Bifurcation Analysis of the 1d and 2d Generalized Swift–hohenberg Equation

Abstract

In this paper, bifurcation of the generalized Swift-Hohenberg equation is considered. We first study the bifurcation of the generalized Swift-Hohenberg equation in one spatial dimension with three kinds of boundary conditions. With the help of Liapunov-Schmidt reduction, the original equation is transformed to the reduced system, and then the bifurcation analysis is carried out. Secondly, bifurcation of the generalized Swift-Hohenberg equation in two spatial dimensions with periodic boundary conditions is also considered, using the perturbation method, asymptotic expressions of the nontrivial solutions bifurcated from the trivial solution are obtained. Moreover, the stability of the bifurcated solutions is discussed.

Bifurcation analysis of a modified Swift–Hohenberg equation

“…Compared with our previous work [18], in which β|∇u| 2 in (1.3) is replaced by βu 2 , in the case (n, m) = (k, k), the type of the bifurcation points and the stability of the four bifurcated solutions don't change, despite γ (0) and the eigenvalue λ(ε) of the linearized eigenvalue problem around the nontrivial solutions are different from the ones in [18].…”

“…In our previous work [18], we consider the bifurcation of the two-dimensional generalized Swift-Hohenberg equation In [19], the existence of a global attractor is proven for the modified Swift-Hohenberg equation which reads ∂u ∂t = αu − (1 + ∆) 2 u + β|∇u| 2 − u 3 , (1.2) near the onset of instability with the Dirichlet's boundary condition and periodic boundary condition, where α is a bifurcation parameter, and β is a constant. The term β|∇u| 2 reminds us of the Kuramoto-Sivashinsky equation, introduced by Kuramoto [20] as a model describing phase turbulence in reaction-diffusion systems and by Sivashinsky [21] for the study of flame propagation.…”

“…We will compare the results in this paper and in [18], and show the impact of different nonlinear terms βu 2 − u 3 and β|∇u| 2 − u 3 on the various types of the bifurcation points, and the impact on the stability of the bifurcated solutions.…”

Dynamical Bifurcation of the One Dimensional Modified Swift-Hohenberg Equation

“…As in the case of the MSHE, it is known from [4,8,11,20] that under the condition (1.4) and the periodic condition, the SHE and the GSHE bifurcate from the trivial states to S 1 -attractorsà N (α) and N (α), respectively. However, there are big differences in the structures of A N (α),à N (α), and N (α).…”

“…In particular, there has been much efforts on the bifurcation analysis as a way of understanding pattern formations. See [4,8,10,11,14,15,19,20] for recent development in this direction.…”