Slow-moving pattern interfaces in general directions for a two-dimensional Swift-Hohenberg-type equation

TL;DR

This paper rigorously proves the existence of slow-moving pattern interfaces with arbitrary directions in a 2D Swift-Hohenberg-type model near Turing instability, advancing understanding of pattern formation mechanisms.

Contribution

It develops a rigorous framework using spatial dynamics and non-standard centre manifold theory to establish solutions describing pattern interfaces in symmetric systems.

Findings

Proves bifurcation of slow-moving interfaces in 2D Swift-Hohenberg models.

Addresses technical challenges like spectral gaps and resonances.

Provides a mathematical foundation for pattern invasion phenomena.

Abstract

We rigorously prove the bifurcation of slow-moving pattern interfaces with general direction in a two-dimensional Swift-Hohenberg-type model close to a Turing instability for a large class of nonlinearities. These interfaces describe the invasion of stripe and hexagonal patterns into the spatially homogeneous state and model a possible mechanism for pattern formation, as observed in a wide range of real-world applications. For this, we develop a rigorous framework to establish the existence of such solutions using spatial dynamics and non-standard centre manifold theory. Our approach exploits geometric and algebraic structures generic to $\mathrm{O}(2)$-symmetric pattern-forming systems near a Turing instability, and addresses fundamental technical challenges due to a non-uniform spectral gap around the imaginary axis, quadratic resonances induced by the hexagonal structure, and the high-dimensional phase space of the reduced equations.