具有分段连续初始条件的Burgers方程的复奇点

中文摘要 AI 辅助

已有大量研究致力于理解非线性偏微分方程（PDE）解的复奇点如何在t=0+时自发产生并在t>0时传播，以及其行为如何影响实轴上的解。尽管小时间极限在这些研究中至关重要，但人们仍未充分理解复奇点在t=0+时的产生机制，包括空间变量非解析的初始条件的情况。本文以Burgers方程作为典型非线性PDE，研究分段光滑初始条件的复平面奇点。通过匹配渐近展开，我们展示了无穷多奇点如何从间断点产生，其模式可通过Lambert-W函数的分支描述。对于各类初始条件，我们观察到这些奇点如何重新排列以契合适当的、经精确描述的长时间行为，包括S形行波、常面积（三角波）相似解和N波解。就Burgers方程而言，我们对分段连续初始条件的奇点传播进行小时间渐近分析，阐明了扩散主导平流时内部区域出现的各类通用行为。更广泛地说，本研究为理解具有非解析初始条件的非线性偏微分方程解的复平面行为迈出了一步。

英文摘要

There is a body of research devoted to understanding how complex singularities of solutions of nonlinear partial differential equations (pdes) spontaneously emerge at $t=0^+$ and propagate for $t>0$, and how their behaviour affects the solution on the real axis. Despite the importance of the small-time limit in these studies, there is still a lack of understanding of how complex singularities are born at $t=0^+$, including for initial conditions that are not analytic functions of the spatial variable. In this paper, we use Burgers' equation as a prototype nonlinear pde and study the complex-plane singularities for initial conditions that are piecewise smooth. Using matched asymptotic expansions, we show how infinitely many singularities emerge from points of discontinuity in a pattern that can be described using branches of the Lambert-$W$ function. For various initial conditions, we observe how these singularities rearrange themselves to align with the appropriate exactly-described long-time behaviour, including sigmoid-shaped travelling waves, constant-area (triangular wave) similarity solutions and $N$-wave solutions. In terms of Burgers' equation, our small-time asymptotic analysis of the singularity propagation for piecewise-continuous initial conditions illustrates the types of generic behaviours that arise for inner regions when diffusion dominates advection. More generally, this work is a step towards understanding complex-plane behaviour of solutions of nonlinear partial differential equations with non-analytic initial conditions.

发表机构

• Queensland University of Technology（昆士兰科技大学）

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