论文标题
Tyger的眼睛：Inviscid Burgers方程中的早期共鸣和奇异性
Eye of the Tyger: early-time resonances and singularities in the inviscid Burgers equation
论文作者
论文摘要
我们为正弦波初始条件以一个空间维度的一个空间维度以一个空间维度的时间域中绘制了一个奇异的景观。到目前为止，这些未发现的复杂奇点以围绕原始时间为中心的眼睛形状排列。有趣的是，由于眼睛沿着想象的时间轴被压扁，因此在第一个真正的奇异性 - 预击之前，复杂的时间奇异性可能在物理上变得有意义。确实，使用$ t = 0 $左右的速度泰勒表示，收敛的损失大约发生在被考虑的单模型和多模型模型的前冲击时间的2/3中。此外，融合的丧失伴随着最初局部谐振行为的出现，正如我们声称的那样，这是所谓的Tyger现象的时间表现，在Galerkin截断的无污液的实现中报道了[Ray等人，物理学。 Rev. E 84，016301（2011）]。我们以两种互补和独立的手段来支持我们的早期泰格人的发现，即通过对速度的Time-Taylor系列的渐近分析，以及采用Lagrangian坐标的新型奇异理论。 最后，我们采用两种方法来减少早期泰格（Tygers）的幅度，一种是泰格（Tyger）清除，它从速度中去除了大型傅立叶模式，并且是文献中已知的过程的一种变体。另一种方法实现了一个迭代的紫外线完成，最有趣的是，一旦泰勒级数以速度差异，它将迭代地恢复能量的保存。我们的技术直接适应更高的维度和/或应用于其他流体动力学方程。
We chart a singular landscape in the temporal domain of the inviscid Burgers equation in one space dimension for sine-wave initial conditions. These so far undetected complex singularities are arranged in an eye shape centered around the origin in time. Interestingly, since the eye is squashed along the imaginary time axis, complex-time singularities can become physically relevant at times well before the first real singularity -- the pre-shock. Indeed, employing a time-Taylor representation for the velocity around $t=0$, loss of convergence occurs roughly at 2/3 of the pre-shock time for the considered single- and multi-mode models. Furthermore, the loss of convergence is accompanied by the appearance of initially localized resonant behaviour which, as we claim, is a temporal manifestation of the so-called tyger phenomenon, reported in Galerkin-truncated implementations of inviscid fluids [Ray et al., Phys. Rev. E 84, 016301 (2011)]. We support our findings of early-time tygers with two complementary and independent means, namely by an asymptotic analysis of the time-Taylor series for the velocity, as well as by a novel singularity theory that employs Lagrangian coordinates. Finally, we apply two methods that reduce the amplitude of early-time tygers, one is tyger purging which removes large Fourier modes from the velocity, and is a variant of a procedure known in the literature. The other method realizes an iterative UV completion, which, most interestingly, iteratively restores the conservation of energy once the Taylor series for the velocity diverges. Our techniques are straightforwardly adapted to higher dimensions and/or applied to other equations of hydrodynamics.
