Shedding new light on the robustness of periodic patterns: wavenumber selection laws via nonlinear stability of pushed fronts and source defects

• Applicant: Dr. Björn de Rijk
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 491897824

Invasion into unstable states plays a crucial role in describing the initial formation of complex coherent structures in many physical systems. In large spatial domains, localized perturbations to the unstable state typically grow, saturate at finite amplitude, and spread into the unstable state, creating a new spatially periodic pattern in the wake of a propagating invasion front. A fundamental question is then to predict both the speed of this invasion front as well as which pattern it selects in its wake. The marginal stability conjecture asserts that speeds and wavenumbers selected by propagation of localized disturbances are determined by the distinguished front solutions which are marginally spectrally stable in an appropriate sense. Our goal is to prove this marginal stability conjecture for pattern selection, which has been an open problem since its formulation in the 1980s. To achieve this goal, we establish a sharp nonlinear stability theory for pattern-forming fronts whose propagation is driven by a localized mode at the front interface. Our theory demonstrates that these so-called pushed fronts attract initial data supported on a half-line, and therefore determine both propagation speeds and selected wave numbers for invasion from highly localized initial data. On the technical level, we use far-field/core methods to capture the challenging interaction between the neutral translational eigenmode driving the propagation and the outgoing diffusive mode associated with the periodic pattern in the wake of the front. The adjustment of the front interface in response to excitement of the translational eigenmode creates a nonlocalized phase mixing problem for the pattern in the wake, which we accommodate with the aid of a phase modulation ansatz and a framework based on spaces of bounded, uniformly continuous functions. For dispersive-dissipative pattern-forming systems such as the Lugiato-Lefever equation, which are incompatible with such function spaces, we develop alternative functional frameworks, based on modulation spaces. The methods we develop are generally useful in any setting involving the interaction of localized modes and outward diffusive transport. Specifically, we aim to employ our techniques to significantly advance the nonlinear stability theory of source defects, which act as organizers of the dynamics far from an equilibrium or rest state by emitting distinguished periodic patterns from a localized core. By proving that source defects are nonlinearly stable against fully nonlocalized perturbations, we lift all previous assumptions on localization of perturbations and demonstrate that source defects are more robust pattern-selection mechanisms than always assumed.

• DFG Programme: Research Grants
• International Connection: USA
• Cooperation Partner: Montie Avery, Ph.D.
• Co-Investigator: Professor Dr. Guido Schneider