发表机构
Mathematical Institute, University of Oxford; School of Mathematics and Statistics, The University of Sydney（牛津大学数学研究所; 悉尼大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究通过辅助线性PDE和阶乘-幂分析，揭示了脉冲启动的KdV5方程中快速振荡源于Stokes现象，并验证了复杂Stokes结构的存在。
AI 中文摘要
我们考虑了在脉冲启动的五阶Korteweg--de Vries方程（KdV5）中快速传播、指数小振幅振荡的出现，并证明这些波源于小时间边界层中的Stokes现象。我们在时间上重新缩放问题，构造了一个辅助线性偏微分方程（lKdV5），该方程再现了相关的指数渐近结构，并通过两种主要方法进行研究。首先，利用最速下降法分析辅助偏微分方程的傅里叶变换解，以提取相关的渐近贡献和Stokes曲线的位置。其次，我们通过直接的阶乘-幂分析重现这些结果。在此分析中，我们利用初始和远场条件，避免了指数渐近方法通常需要的局部渐近匹配，而这种匹配对于非线性偏微分方程通常是不可行的。随后，将类似的阶乘-幂分析应用于完整的非线性偏微分方程，该方程无法使用最速下降法研究。结果与数值模拟吻合良好，展示了非衰减振荡的快速传播和振幅在远场衰减的瞬态波纹，并揭示了控制其出现的复杂Stokes结构（包括正常和高阶Stokes曲线）。这种在时间边界层中的复杂Stokes结构是一种新颖现象。
英文摘要
We consider the emergence of rapidly propagating, exponentially small-amplitude oscillations in the impulsively started fifth-order Korteweg--de Vries equation (KdV5), and demonstrate that these waves originate due to Stokes' phenomenon in a small-time boundary layer. We rescale the problem in time and construct an auxiliary linear PDE (lKdV5) that reproduces the relevant exponential asymptotic structure, and proceed by two main methods. Firstly, the Fourier transform solution of the auxiliary PDE is analysed via the method of steepest descents to extract the relevant asymptotic contributions and the locations of the Stokes curves. Secondly, we reproduce these results using a direct factorial-over-power analysis. In this analysis, we make use of the initial and far-field conditions to avoid performing the local asymptotic matching that is typically required by exponential asymptotic methods and is frequently impossible for nonlinear PDEs. A similar factorial-over-power analysis is then applied to the full nonlinear PDE, which cannot be studied using steepest descent methods. The results agree well with numerical simulations, demonstrating both the rapid propagation of non-decaying oscillations and transient ripples whose amplitudes decay in the far field, and revealing the complex Stokes structure (including both normal and higher-order Stokes curves) that governs their emergence. Such a complex Stokes structure in a temporal boundary layer is a novel phenomenon.
Comments35 pages, 7 figures
