论文标题
在中性稳定性的一维代数生长波的二维扩展上
On the two-dimensional extension of one-dimensional algebraically growing waves at neutral stability
论文作者
论文摘要
这项工作考虑了两个线性运算符，它们产生了被归类为中性稳定的波浪模式，但响应的响应会随着时间的推移而增长或衰减。以前，金等人。 （Phys。Rev。Fluids，1，2016，073604：1-19）和Huber等人。 （Ima J.Appl。Math。，85，2020，309-340）检查了这些操作员控制的一维（1D）波传播。在这里，我们将线性操作员扩展到两个空间维度（2D）并检查所得的解决方案。我们发现，维度的增加导致长期行为，其中从1D解决方案中缩小了幅度降低$ t^{\ frac {-1} {2}} $的因子。因此，在1D中以$ t^{\ frac {1} {2}} $在1D中以$ t^{\ frac {1}} $为单位的溶液区域在2D中是代数中性的，而在1D中，在1D中衰减（代数或指数性地）在2D中更快地衰减。此外，我们发现这两个线性运算符接纳了与空间和时间相同相似性变量的函数的长期解决方案。
This work considers two linear operators which yield wave modes that are classified as neutrally stable, yet have responses that grow or decay in time. Previously, King et al. (Phys. Rev. Fluids, 1, 2016, 073604:1-19) and Huber et al. (IMA J. Appl. Math., 85, 2020, 309-340) examined the one-dimensional (1D) wave propagation governed by these operators. Here, we extend the linear operators to two spatial dimensions (2D) and examine the resulting solutions. We find that the increase of dimension leads to long-time behaviour where the magnitude is reduced by a factor of $t^{\frac{-1}{2}}$ from the 1D solutions. Thus, regions of the solution which grew algebraically as $t^{\frac{1}{2}}$ in 1D now are algebraically neutral in 2D, whereas regions which decay (algebraically or exponentially) in 1D now decay more quickly in 2D. Additionally, we find that these two linear operators admit long-time solutions that are functions of the same similarity variable that contracts space and time.
