General stability theory for 3D localized patterns

Develop a general framework to determine spectral and nonlinear stability of three‑dimensional localized patterns—including (1+2)D fronts, (2+1)D planar patches, and fully localized 3D structures—in models such as the Swift–Hohenberg equation, reaction–diffusion systems, and hydrodynamic PDEs.

Background

While existence and bifurcation structures of many 3D localized patterns are being uncovered numerically, systematic stability theory in three dimensions is lacking.

A unified approach combining variational methods and spatial‑dynamical ideas is anticipated to be necessary, particularly in the presence of conserved quantities and complex spectra.