Shedding new light on the robustness of periodic patterns: nonlinear stability of modulated wave trains in a pure L^infty-framework

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

Our goal is to develop a new mathematical framework to study the nonlinear stability and modulational behavior of periodic waves in pattern-forming systems on spatially extended domains. In contrast to state-of-the-art methods, our proposed framework does not rely on localization or periodicity properties of perturbations or modulations. Instead, we only require that the initial perturbation and sufficiently many of its derivatives are bounded, i.e., they are small in the supremum norm. Thus, we expect to establish nonlinear stability for much larger classes of modulational initial data than those being dealt with in the current literature. In particular, this would allow us to systematically handle nonlocalized modulations of the wavenumber, which has been an open problem for the last decades. Our anticipated results imply that modulated periodic patterns are more robust than previously assumed as localization or periodicity requirements on perturbations, as present in all current literature, can be completely lifted.Closing the nonlinear argument in a pure L^infty-framework is challenging, since the linearization about the periodic wave has continuous spectrum touching the origin in a quadratic tangency due to translational invariance. Thus, on the linear level, the translational mode is diffusive and cannot be expected to decay in L^infty-spaces. We will address this difficulty by first decomposing into phase and amplitude variables, where the phase variable accounts for the critical translational mode and the amplitude variable is expected to exhibit higher-order decay. One observes that the phase variable enters the nonlinearity in the perturbation equation as a derivative only. We aim to exploit this observation to close the nonlinear iteration by employing the smoothing properties induced by the critical part of the spectrum, i.e., derivatives of bounded functions decay at an algebraic rate when subject to diffusion. The scalar equation for the critical phase variable takes the form of a perturbed viscous Hamilton-Jacobi equation. In the absence of additional symmetries, its dynamics needs to be controlled using the maximum principle or Cole-Hopf transform. Our ultimate goal is to apply the developed tools to more delicate situations, where there are multiple critical modes, e.g. due to additional conservation laws.

• DFG Programme: Research Grants
• Co-Investigator: Professor Dr. Guido Schneider