A Formal Geometric Blow-up Method for Pattern Forming Systems

TL;DR

This paper develops a formal geometric blow-up method combined with a fast-slow extension of multiple scales to derive modulation equations near bifurcation points in PDEs, applicable to various pattern-forming systems.

Contribution

It introduces a systematic, formal approach that generalizes classical modulation theory using geometric blow-up and fast-slow analysis for PDE bifurcations.

Findings

Derivation of modulation equations for Turing, Hopf, Turing-Hopf, and stationary bifurcations.

Modulation equations have time-dependent coefficients and complex spatial-temporal scales.

The method offers a first step towards rigorous analysis of dynamic bifurcations in PDEs.

Abstract

We extend and apply a recently developed approach to the study of dynamic bifurcations in PDEs based on the geometric blow-up method. We show that this approach, which has so far only been applied to study a dynamic Turing bifurcation in a cubic Swift-Hohenberg equation, can be coupled with a fast-slow extension of the method of multiple scales. This leads to a formal but systematic method, which can be viewed as a fast-slow generalisation of the formal part of classical modulation theory. We demonstrate the utility and versatility of this method by using it to derive modulation equations, i.e. simpler closed form equations which govern the dynamics of the formal approximations near the underlying bifurcation point, in the context of model equations with dynamic bifurcations of (i) Turing, (ii) Hopf, (iii) Turing-Hopf, and (iv) stationary long-wave type. The modulation equations have a familiar form: They are of real Ginzburg-Landau (GL), complex GL, coupled complex GL and Cahn-Hilliard type respectively. In contrast to the modulation equations derived in classical modulation theory, however, they have time-dependent coefficients induced by the slow parameter drift, they depend on spatial and temporal scales which scale in a dependent and non-trivial way, and the geometry of the space in which they are posed is non-trivial due to the blow-up transformation. The formal derivation of the modulation equations provides the first steps toward the rigorous treatment of these challenging problems, which remains for future work.