Spatially extended dislocations produced by the dispersive Swift-Hohenberg equation
Spatially extended dislocations produced by the dispersive Swift-Hohenberg equation
复制标题
由色散 Swift-Hohenberg 方程产生的空间扩展位错
DOI：
10.1103/physreve.107.044214
复制
发表时间：
2023
影响因子：
2.4
通讯作者：
Bradley, R. Mark
中科院分区：
文献类型：
--
作者：
Balch, Brenden;Shipman, Patrick D.;Bradley, R. Mark
前人的研究结果表明，在二维Kuramoto-Sivashinsky方程中加入一个线性色散项对图案的形成有显著的影响，因此我们研究了在二维Kuramoto-Sivashinsky方程中加入一个线性色散项的Swift-Hohenberg方程，即色散Swift-Hohenberg方程（DSHE）。DSHE产生带有空间扩展缺陷的条纹图案，我们称之为接缝。一个缝被定义为一种错位，它沿着一条相对于反射对称轴线倾斜的线段被涂抹出来。与色散Kuramoto-Sivashinsky方程相反，DSHE具有接近不稳定阈值的不稳定波长窄带。这允许进行分析进展。我们证明了DSHE接近阈值的振幅方程是各向异性复金兹堡-朗道方程（ACGLE）的一种特殊情况，DSHE中的接缝对应于ACGLE中的螺旋波。煤层缺陷和相应的螺旋波倾向于组织成链状，我们得到了螺旋波核的速度和它们之间的间距的公式。在强色散的极限下，微扰分析得出了条纹图案的振幅和波长与其传播速度之间的关系。ACGLE和DSHE的数值积分证实了这些分析结果。
Motivated by previous results showing that the addition of a linear dispersive term to the two-dimensional Kuramoto-Sivashinsky equation has a dramatic effect on the pattern formation, we study the Swift-Hohenberg equation with an added linear dispersive term, the dispersive Swift-Hohenberg equation (DSHE). The DSHE produces stripe patterns with spatially extended defects that we call seams. A seam is defined to be a dislocation that is smeared out along a line segment that is obliquely oriented relative to an axis of reflectional symmetry. In contrast to the dispersive Kuramoto-Sivashinsky equation, the DSHE has a narrow band of unstable wavelengths close to an instability threshold. This allows for analytical progress to be made. We show that the amplitude equation for the DSHE close to threshold is a special case of the anisotropic complex Ginzburg-Landau equation (ACGLE) and that seams in the DSHE correspond to spiral waves in the ACGLE. Seam defects and the corresponding spiral waves tend to organize themselves into chains, and we obtain formulas for the velocity of the spiral wave cores and for the spacing between them. In the limit of strong dispersion, a perturbative analysis yields a relationship between the amplitude and wavelength of a stripe pattern and its propagation velocity. Numerical integrations of the ACGLE and the DSHE confirm these analytical results.