Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system

TL;DR

This paper investigates pattern formation in a nonlinear advection system, identifying solutions related to convective and absolute instabilities, and explores their stability and sensitivity to boundary conditions using geometric singular perturbation theory and numerical methods.

Contribution

It introduces a novel analysis of stationary fronts in a nonlinear pattern-forming system as a slow-fast spatial dynamical system, clarifying the role of canard trajectories and boundary conditions.

Findings

Identification of two types of stationary fronts linked to convective and absolute instability.

Quantitative understanding of noise-sustained structures in the system.

Sensitivity of front location to upstream boundary conditions and statistical properties under stochastic inputs.

Abstract

A subcritical pattern-forming system with nonlinear advection in a bounded domain is recast as a slow-fast system in space and studied using a combination of geometric singular perturbation theory and numerical continuation. Two types of solutions describing the possible location of stationary fronts are identified, whose origin is traced to the onset of convective and absolute instability when the system is unbounded. The former are present only for nonzero upstream boundary conditions and provide a quantitative understanding of noise-sustained structures in systems of this type. The latter correspond to the onset of a global mode and are present even with zero upstream boundary condition. The role of canard trajectories in the nonlinear transition between these states is clarified and the stability properties of the resulting spatial structures are determined. Front location in the convective regime is highly sensitive to the upstream boundary condition and its dependence on this boundary condition is studied using a combination of numerical continuation and Monte Carlo simulations of the partial differential equation. Statistical properties of the system subjected to random or stochastic boundary conditions at the inlet are interpreted using the deterministic slow-fast spatial-dynamical system.