Recurrent traveling waves in a two-dimensional saw-toothed cylinder and their average speed
Recurrent traveling waves in a two-dimensional saw-toothed cylinder and their average speed
复制标题
DOI：
10.1016/j.jde.2013.07.038
复制
发表时间：
2013-11
影响因子：
2.4
通讯作者：
B. Lou;H. Matano;Ken-Ichi Nakamura
中科院分区：
文献类型：
--
作者：
B. Lou;H. Matano;Ken-Ichi Nakamura
研究了具有起伏边界的二维无限长圆柱中平面曲线的曲率相关运动。运动定律由V= κ+ A给出，其中V是曲线的法向速度，κ是曲率，A是正常数。边界波动被假定为几乎周期性的，或者更一般地说，在某种意义上是经常性的。首先给出了常返行波的定义，并建立了常返行波存在的充分必要条件。然后，我们证明了行波是渐近稳定的，如果它存在。接下来，我们表明，一个规则的行波有一个明确的平均速度，如果边界形状是严格遍历的。最后，我们研究了所谓的“虚钉扎”，这意味着行波以零平均速度在整个圆柱上传播。这种特殊的情况只会发生在非周期性的环境中，如果边界波动是周期性的，则不会发生。
We study a curvature-dependent motion of plane curves in a two-dimensional infinite cylinder with spatially undulating boundary. The law of motion is given by V= κ+ A, where V is the normal velocity of the curve, κ is the curvature, and A is a positive constant. The boundary undulation is assumed to be almost periodic, or, more generally, recurrent in a certain sense. We first introduce the definition of recurrent traveling waves and establish a necessary and sufficient condition for the existence of such traveling waves. We then show that the traveling wave is asymptotically stable if it exists. Next we show that a regular traveling wave has a well-defined average speed if the boundary shape is strictly ergodic. Finally we study what we call “virtual pinning”, which means that the traveling wave propagates over the entire cylinder with zero average speed. Such a peculiar situation can occur only in non-periodic environments and never occurs if the boundary undulation is periodic.