Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom.
Centre of Mathematical Modelling and Simulation, Institut Teknologi Bandung, 1st Floor, Labtek III, Jl. Ganesha No. 10, Bandung 40132, Indonesia.
Phys Rev E. 2017 Dec;96(6-1):062214. doi: 10.1103/PhysRevE.96.062214. Epub 2017 Dec 21.
We consider the discrete Swift-Hohenberg equation with cubic and quintic nonlinearity, obtained from discretizing the spatial derivatives of the Swift-Hohenberg equation using central finite differences. We investigate the discretization effect on the bifurcation behavior, where we identify three regions of the coupling parameter, i.e., strong, weak, and intermediate coupling. Within the regions, the discrete Swift-Hohenberg equation behaves either similarly or differently from the continuum limit. In the intermediate coupling region, multiple Maxwell points can occur for the periodic solutions and may cause irregular snaking and isolas. Numerical continuation is used to obtain and analyze localized and periodic solutions for each case. Theoretical analysis for the snaking and stability of the corresponding solutions is provided in the weak coupling region.
我们考虑了离散的 Swift-Hohenberg 方程，其中包含立方和五次方非线性项，这是通过对 Swift-Hohenberg 方程的空间导数进行中心有限差分离散得到的。我们研究了离散化对分岔行为的影响，确定了耦合参数的三个区域，即强耦合、弱耦合和中间耦合。在这些区域内，离散的 Swift-Hohenberg 方程的行为与连续极限相似或不同。在中间耦合区域，对于周期性解可能会出现多个 Maxwell 点，这可能会导致不规则的蛇形运动和孤立子。数值延拓用于获得和分析每种情况下的局域和周期性解。在弱耦合区域提供了对相应解的蛇形运动和稳定性的理论分析。
