Damping filter method for obtaining spatially localized solutions
Damping filter method for obtaining spatially localized solutions
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用于获得空间局部解的阻尼滤波器方法
DOI：
10.1103/physreve.89.052910
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发表时间：
2014
影响因子：
2.4
通讯作者：
Toshiki Teramura and Sadayoshi Toh
中科院分区：
文献类型：
--
作者：
Toshiki Teramura and Sadayoshi Toh;Toshiki Teramura and Sadayoshi Toh
空间局域结构是湍流和其他时空混沌系统的关键组成部分。从动力系统的观点来看，得到相应的精确解是可取的，尽管它们的存在性不能保证。介绍了一种阻尼滤波方法，可得到不同的局部解，并适用于两个典型实例。该方法引入了空间选择性阻尼效应，可以很好地猜测精确解，并通过对阻尼幅值的延拓得到精确解。第一个目标是Swift-Hohenberg方程的稳定解，它是局域解共存的双稳系统的代表，是展向局域情况的一个模型。不仅找到了众所周知的蛇形分支的解，而且还找到了孤立分支的解，即“孤立”分支的解在相空间中具有随阻尼幅值扩展的延拓路径。这表明这种空间选择性激励机制在寻找空间定域解方面具有优势。第二个目标是Kuramoto-Sivashinsky方程的空间局部化行波解，这是一个流向局部化情况的模型。由于空间选择性阻尼效应打破了伽利略和平移不变性，当阻尼有效时，传播速度不能唯一确定，当这些不变性恢复时，就会出现奇点。我们证明了通过施加一个简单的条件可以避免这种奇点，并在特定的传播速度下获得了局域行波解。
Spatially localized structures are key components of turbulence and other spatiotemporally chaotic systems. From a dynamical systems viewpoint, it is desirable to obtain corresponding exact solutions, though their existence is not guaranteed. A damping filter method is introduced to obtain variously localized solutions and adapted in two typical cases. This method introduces a spatially selective damping effect to make a good guess at the exact solution, and we can obtain an exact solution through a continuation with the damping amplitude. The first target is a steady solution to the Swift-Hohenberg equation, which is a representative of bistable systems in which localized solutions coexist and a model for spanwise-localized cases. Not only solutions belonging to the well-known snaking branches but also those belonging to isolated branches known as “isolas” are found with continuation paths between them in phase space extended with the damping amplitude. This indicates that this spatially selective excitation mechanism has an advantage in searching spatially localized solutions. The second target is a spatially localized traveling-wave solution to the Kuramoto-Sivashinsky equation, which is a model for streamwise-localized cases. Since the spatially selective damping effect breaks Galilean and translational invariances, the propagation velocity cannot be determined uniquely while the damping is active, and a singularity arises when these invariances are recovered. We demonstrate that this singularity can be avoided by imposing a simple condition, and a localized traveling-wave solution is obtained with a specific propagation speed.