Rigorous convergence of truncated-domain farfield–core computations to unbounded-domain fronts

Establish rigorous convergence results that connect solutions of the truncated periodic-domain farfield–core boundary value problem used to numerically approximate modulated invasion fronts in the cubic Swift–Hohenberg equation ∂t u = −(1+∂x^2)^2 u + μ u − u^3 to bona fide front solutions on the unbounded spatial domain, including quantitative error estimates as the domain size tends to infinity and conditions on boundary treatments.

Background

The authors compute pulled invasion fronts via a farfield–core decomposition posed on a large but finite periodic domain in the co-moving spatial variable. This approach leverages expected spatial localization and exponential decay to justify truncation and periodic boundary conditions for efficient spectral discretization.

While numerical evidence suggests the truncation error is exponentially small in the domain size, the authors emphasize that a rigorous convergence theory linking truncated-domain solutions to true unbounded-domain front solutions is currently lacking, noting only preliminary progress. This creates a well-defined theoretical problem concerning existence, convergence rates, and parameter dependence of the approximation scheme.