Existence boundary of Turing patterns at larger μ

Characterize the existence boundary of stationary wavetrains (Turing patterns) u(x) = u_p(kx) in the cubic Swift–Hohenberg equation ∂t u = −(1+Δ)^2 u + μ u − u^3 for non-small μ by determining the full bifurcation structure and delimiting curves in (k, μ)-parameter space, thereby fully describing the intricate existence boundary observed numerically.

Background

Beyond the small-μ regime, the family of stationary wavetrains exhibits complex existence and bifurcation structures, with intricate boundaries in wavenumber–parameter space. The authors provide numerical evidence of such complexity, including at μ ≈ 0.9, prior to additional equilibria bifurcations.

They explicitly note that this existence boundary has not been fully explored, indicating the need for a comprehensive analysis and continuation of wavetrain solution branches and boundaries to map the full parameter space beyond weakly nonlinear predictions.