Numerical Continuation and Bifurcation in Nonlinear PDEs: Stability, invasion and wavetrains in the Swift-Hohenberg equation

TL;DR

The paper addresses how to compute and understand wavetrain existence, stability, and invasion fronts in the Swift–Hohenberg PDE using numerical continuation. It integrates Floquet–Bloch spectral analysis, Eckhaus and zigzag stability boundaries, weighted-space dispersion, and a far-field–core decomposition to characterize pulled fronts, all implemented with Newton-based continuation and flexible discretizations. Key contributions include explicit formulations of wavetrain existence and stability boundaries in 1D and 2D, algorithms for front invasion speeds via double-root/pinching analysis, and practical, shareable code suitable for extension to reaction–diffusion systems and higher dimensions. The work advances the practical toolkit for pattern formation research by providing an accessible, tutorial-style framework for computing coherent structures and their dynamics in PDEs.

Abstract

We discuss some aspects of numerical continuation and bifurcation for partial differential equations, specifically pattern formation and coherent structures. For the sake of clarity we focus on wavetrains, stability and associated invasion processes in the paradigmatic cubic Swift-Hohenberg equation (SHE). We do not aim at a review of numerical continuation for PDE or pattern formation in SHE in any generality, rather our goal is to provide an entry point for interested students and colleagues to the application of continuation methods. We provide access to numerical implementations and hope that our presentation provides an introductory guideline that can also be used for teaching.

Figures (6)

• Figure 1: Space-time diagram of pattern-forming invasion front in the Swift-Hohenberg equation \ref{['e:SHeqn']} with $\mu = 0.25$ and localized initial condition $u(x,0) = \frac{1}{2} e^{-x^2}$; color coding for $u(x,t)$. The computation was done with 4th order exponential time-differencing and spectral discretization in space. The underlying only partially shown computational domain is $x\in[-80\pi,80\pi]$ with $dt = 0.01$ and $N = 2^{13}$ Fourier modes. A stable wavetrain (upper left area) invades the homogeneous unstable state, cf. Figures \ref{['f:wt-existence']} and \ref{['f:SHE-Busse']}; time-slices near an invasion front are shown in Figure \ref{['fig:sh-front']}.
• Figure 2: Numerical continuation of wavetrains in \ref{['e:SHeqn']} implemented in AUTO auto. (a) Bifurcation diagram for $\mu=0.1$ (blue) and $\mu=0.9$ (black) in terms of the wavenumber $k$; (b) solution profiles at marked locations in (a).
• Figure 3: (a) Floquet-Bloch spectrum near $\sigma=0$ for fixed $\mu=0.1$ and one value of $k$ in the stable region, another beyond the Eckhaus boundary as marked by black bullets in (b). (b) Stability boundaries with unstable region shaded. For larger $\mu$ the $k$-values of the Eckhaus boundary increase further, in particular beyond $k=1$.
• Figure 4: Parts of the essential spectrum of selected $u_p(k;\mu)$ computed via Floquet-Bloch operator. The direct computation via an eigenvalue solver is in this case faster than the computation by continuation in $\sigma$. Parameters lie on the Eckhaus boundary: For panels (a,b,c) $\mu\approx 10$, $k\approx 0.9$ just outside the range in Figure \ref{['f:SHE-Busse']}; for panel (d) $\mu\approx 5$, $k\approx 0.8$ in the zigzag unstable regime. In panels (a,b) $\sigma\in[-1,1]$, $\tau=0$, and we plot the two, respectively four, most unstable eigenvalues, illustrating the band structure of the spectrum with disconnected intervals. In panels (c,d) we plot only the most unstable eigenvalue for $\sigma\in[-0.5,0.5]$ and $\tau=j/10$ for $j=1,\ldots 50$.
• Figure 5: The above figure depicts the unweighted essential spectrum $\Sigma_{\mu,c_*}$ (red), the weighted essential spectrum $\Sigma_{\mu,c_*,\eta}$ for weight with $\mathrm{Re}\, \nu_*(c_*)<\eta<0$ (orange) and with $\eta = \mathrm{Re}\,\nu_*(c_*)$ (green), and absolute spectrum $\Sigma_{\mu,c_*}^\mathrm{abs}(0)$ (blue), defined below, of the homogeneous state $u = 0$ at the linear spreading speed $c = c_*$ and $\mu = 1/4$. Insets plot the spatial eigenvalues $\{\nu_j\}$ for $\lambda$ values, with square dots denoting a double root. The dashed line in upper left inset denotes the shift caused by weight $\eta$ in orange case.