标题： 超越实爆破：Masuda绕道和复数全纯性 显示英文标题

标题： Beyond real blow-up: Masuda detours and complex holonomy

摘要： 对于实数$\mathbf{b}$，考虑类似\begin{equation*} \mathbf{w}_t=\mathbf{w}_{\boldsymbol{\xi}\boldsymbol{\xi}} + \mathbf{b}(\boldsymbol{\xi})\,\mathbf{w}^2 \end{equation*}的二次热方程，在$\boldsymbol{\xi}\in(0,\pi)$上带有Neumann边界条件。对于$\mathbf{b}$=1，Kyûya Masuda 在20世纪80年代的开创性工作旨在通过穿越复时间的绕行方法来避免PDE爆破，这在有限实时间内发生。对前两个Galerkin模态$\mathbf{w}=x+y \cos\boldsymbol{\xi}$进行直观投影会引导我们得到一个ODE的简略模型。 如在PDE中，从$x_0$开始的时空齐次解$y=0\neq x\in\mathbb{R}$在有限实时间$t=T=1/x_0$处爆破。 秉承Masuda的精神，我们将实解析ODE解扩展到复时间，并扩展到实四维$(x,y)\in\mathbb{C}^2$，以绕过在$t=T$处的实爆破奇点。 因此，我们研究一般多项式ODE在$(x,y)\in\mathbb{C}^2$上的复叶层，在如$u=1/x,\ z=y/x$的射影紧化中，包括在爆破点的单值性$u=0$。 我们在Poincaré和Siegel型爆破平衡点处获得线性化，基于谱非共振。 我们讨论有理周期非共振以及Diophantine类型的无理准周期非共振对ODE形象中迭代Masuda迂回路径的影响。 最后，我们对全局方面、偏微分方程、离散化和其他应用进行一些评论。 显示英文摘要

摘要： For real $\mathbf{b}$, consider quadratic heat equations like \begin{equation*} \mathbf{w}_t=\mathbf{w}_{\boldsymbol{\xi}\boldsymbol{\xi}} + \mathbf{b}(\boldsymbol{\xi})\,\mathbf{w}^2 \end{equation*} on $\boldsymbol{\xi}\in(0,\pi)$ with Neumann boundary conditions. For $\mathbf{b}$=1, pioneering work by Ky\^uya Masuda in the 1980s aimed to circumvent PDE blow-up, which occurs in finite real time, by a detour which ventures through complex time. Naive projection onto the first two Galerkin modes $\mathbf{w}=x+y \cos\boldsymbol{\xi}$ leads us to an ODE caricature. As in the PDE, spatially homogeneous solutions $y=0\neq x\in\mathbb{R}$ starting at $x_0$ blow up at finite real time $t=T=1/x_0$. In the spirit of Masuda, we extend real analytic ODE solutions to complex time, and to real 4-dimensional $(x,y)\in\mathbb{C}^2$, to circumvent the real blow-up singularity at $t=T$. We therefore study complex foliations of general polynomial ODEs for $(x,y)\in\mathbb{C}^2$, in projective compactifications like $u=1/x,\ z=y/x$, including their holonomy at blow-up $u=0$. We obtain linearizations, at blow-up equilibria of Poincar\'e and Siegel type, based on spectral nonresonance. We discuss the consequences of rational periodic nonresonance, and of irrational quasiperiodic nonresonance of Diophantine type, for iterated Masuda detours in the ODE caricature. We conclude with some comments on global aspects, PDEs, discretizations, and other applications.