Nonlinear waves and pattern-formation in advective systems

• Applicant: Dr. Bastian Hilder
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 543917644

The aim of this project is to study the dynamics of nonlinear waves in pattern-forming systems in the presence of advection. This class of systems has many important applications in biology and hydrodynamics, where advective effects have been identified as a key driving factor. Examples include chemomechanical effects in active fluids and the transition to turbulence in shear flows. In recent years, many minimal models have been proposed to understand the dynamics of these complex systems. These models can be validated against experimental observations using numerical methods. A particularly important class of such models are reaction-diffusion-advection systems, which have recently received much attention in mathematical research. The dynamics of these minimal models can be extremely complex and difficult to understand analytically. However, it is very common in applications that the physical system has a multiscale structure, which is reflected in the minimal model. In this project, we are interested in understanding nonlinear waves and patterns in these models by exploiting their multiscale structure. Specifically, we are interested in: (1) modulated patterns triggered by a slow spatial change in parameters, (2) dynamics close to a resonant instability, (3) time-periodic patterns arising from an oscillator long-wave instability, and (4) identifying structures where the numerically validated minimal models can be formally derived from a physical model. The common mathematical core of these problems is the need to understand the dynamics of an infinite-dimensional problem with a multiscale structure. The methodology is based on well-established mathematical techniques for the treatment of multiscale problems such as center manifolds, modulation equations, and geometric singular perturbation theory. However, applications where structural properties of the problem are broken or generalized typically require mathematical methods beyond the state of the art. In particular, this includes applications with advective effects, slowly drifting parameter multiscale dynamics, and infinite-dimensional fast-slow structures, which are at the core of this project. We will develop new and extend existing techniques for these problems and thus contribute significantly to the development of a comprehensive mathematical framework for infinite-dimensional multiscale systems.

• DFG Programme: WBP Position