Beyond real blow-up: Masuda detours and complex holonomy

TL;DR

This paper explores complex time detours for quadratic heat equations and their ODE caricatures to understand and potentially circumvent finite-time blow-up phenomena, extending Masuda's approach through complex foliations and holonomy analysis.

Contribution

It introduces a novel complex foliation framework for polynomial ODEs related to heat equations, analyzing holonomy and nonresonance conditions to extend solutions beyond blow-up times.

Findings

Complex foliations help analyze blow-up behavior.

Nonresonance conditions influence solution extension.

Holonomy analysis provides insights into PDE and ODE dynamics.

Abstract

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 $t=T=1/x_0$. We aim for ODE "linearization at infinity". Since iterated complex time loops are not feasible, for parabolic PDEs, our PDE-motivated approach is currently limited to ODEs. On the other hand, all ODE results of the present paper exactly embed into certain PDEs of parabolic type, which possess a PDE-invariant Galerkin subspace. 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.