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

Our new preprint on planar pattern interfaces is out! Together with Bastian Hilder (TU Munich), I have continued our research on pattern formation in systems close to a Turing instability. The formation of regular spatial structures — hexagonal patterns, stripes, spots — is one of the most striking phenomena in nature, appearing in fluid convection, vegetation landscapes, chemical reactions, and biological systems alike. A key question is not just which patterns form, but how they spread: typically, a moving front invades an unstable homogeneous state and leaves a patterned state in its wake.

In this new preprint, we study these invasion fronts rigorously in a two-dimensional Swift–Hohenberg-type equation, a prototypical model pattern formation driven by a Turing instability. Specifically, we construct slow-moving planar interfaces that travel with a speed proportional to the amplitude of the pattern they create. This slow-speed scaling reflects the physically relevant regime: close to the onset of instability, a marginal stability analysis predicts that fronts with exactly this speed are selected, meaning they emerge robustly from generic initial conditions. This makes them the natural candidates for explaining how patterns invade space in real experiments.

A distinctive feature of our work is that we treat interfaces travelling in general directions — not just along the natural axes of the pattern lattice. The direction of propagation turns out to matter: the structure of the equations depends sensitively on the angle of the front. To handle this, we develop a rigorous framework based on spatial dynamics and a non-standard centre manifold reduction. This framework is designed to handle essentially all rotation-symmetric semilinear pattern-forming systems close to a Turing instability in two spatial dimensions.

The preprint can be found on arXiv https://arxiv.org/abs/2604.09530.