By Dan J. Hill and David J.B. Lloyd
Spatially localized patterns offer fascinating insights into the emergence of organized behaviors, from weather fronts and desert vegetation to stop-and-go traffic waves and the buckling of rocket cylinders (see Figure 1). While existing mathematical tools—such as bifurcation analysis and singular perturbation theory—facilitate the study of one-dimensional (1D) localized patterns, the development of equivalent tools in higher spatial dimensions remains a significant challenge and has been a major research area for the past 20 years.
The establishment of mathematical pattern formation as an area of study is often attributed to Alan Turing and his seminal 1952 paper [10] on general two-component reaction-diffusion systems of the form
\[\mathbf{u}_t = \mathbf{D}\Delta\mathbf{u} + \mathbf{f(u)},\]
where \(\mathbf{u}\in\mathbb{R}^2\), \(\mathbf{D}\) is a diffusion matrix, \(\Delta\) is the Laplacian, and \(\mathbf{f}\) is a nonlinear function. Towards the end of his life, Turing—who was fascinated by hexagonal patterns in daisies and fluid dynamics—derived a scalar partial differential equation (PDE) that closely resembles the now-standard prototypical model for pattern formation:
\[U_t = -(1+\Delta)^2U - \mu U + \nu U^2 - U^3, \tag1\]
where \(U = U(\mathbf{x},t)\in\mathbb{R},\) and \(\mu\) and \(\nu\) are real parameters. This formula, which is known as the Swift-Hohenberg equation (SHE), was derived again 25 years later as a simple model for Rayleigh-Bénard convection [9].
The SHE undergoes a pattern-forming instability at \(\mu=0\) such that for \(\mu<0\), the previously stable trivial state becomes unstable to patterns of the form \(\cos(\mathbf{k}\cdot\mathbf{x})\). Here, the critical wave vector \(\mathbf{k}\) has the same dimension as the spatial coordinate \(\mathbf{x}\) and satisfies the condition \(|\mathbf{k}|=1\), which is a key indicator of the increased difficulty in the study of multidimensional patterns; 1D patterns have just two critical wave numbers (\(k = 1\) and \(k=-1\)), whereas patterns in higher spatial dimensions have a continuum of critical wave vectors. We can thus restrict to a finite number of bifurcating modes for 1D patterns, but not for multidimensional patterns. In the case of domain-covering patterns, researchers often overcome this limitation by restricting to solutions that lie on a lattice with a given symmetry. Unfortunately, such an approach fails for multidimensional localized patterns.
Figure 1. Various localized patterns in different materials. 1a. A ferrofluid. 1b. A buckling cylinder. 1c. Vertically vibrating granular material. 1d. Sun spots. 1e. Vegetation patches in an arid area. 1f. Vertically vibrating fluid. Figure 1a courtesy of Achim Beetz, 1b courtesy of M. Ahmer Wadee, 1c courtesy of Paul Umbanhowar, 1d courtesy of NASA Goddard Space Flight Center, 1e courtesy of Stephan Getzin, and 1f courtesy of [4].
Radial Spatial Dynamics Approach
Localized axisymmetric patterns may be the simplest type of multidimensional localized patterns, given that the solution \(U\) is rotation invariant and only depends on a single radial variable \(r\). In light of radial spatial dynamics theory [8], we can use radial normal form theory to extend Turing’s analysis and explain the emergence of different types of localized radial patterns [5]. Further studies have expanded this approach to three-dimensional (3D) spots [7] and axisymmetric patterns on the surface of a ferrofluid [3] (see Figure 1a). The ferrofluid scenario takes the form of a 3D free-surface problem and represents a significant departure from the simple ordinary differential equation (ODE) structure of the steady axisymmetric SHE. However, many of the necessary analytic tools for the rigorous study of localized axisymmetric patterns in complex problems remain underdeveloped.
What if a solution displays radial localization but is no longer axisymmetric? For example, what if we want to study a pattern of a compact patch of hexagonality that is surrounded by a uniform state (as in Figure 1a)? To do so, we can consider solutions of the SHE in \((1)\) that possess dihedral symmetry \(\mathbb{D}_m\)—i.e., invariance under a rotation of \(2\pi/m\)—so that we can write \(U\) as an angular Fourier expansion:
\[U(r,\theta) = \sum_{n\in\mathbb{Z}} u_{|n|}(r)\cos(mn\theta). \]
Projecting the SHE onto each Fourier mode yields an infinite-dimensional system of radial differential equations that is difficult to analyze with standard techniques. However, we can approximate this system via a truncated Fourier expansion
\[U(r,\theta) = u_0(r) + 2\sum_{n=1}^N u_n(r)\cos(mn\theta), \tag2\]
with truncation order \(N\in\mathbb{N}.\). A 2008 study used this pseudo-spectral approach to numerically obtain and continue various localized dihedral patterns that bifurcate the trivial state at \(\mu=0\), with a focus on localized hexagon (\(\mathbb{D}_{6}\)) and rhombic (\(\mathbb{D}_{2}\)) patterns [6]. In Figure 2, we plot the bifurcation curve for localized hexagons and compare it with an axisymmetric spot [5, 6].
Figure 2. Numerically continued bifurcation curves of localized spots (in blue) and hexagon patches (in yellow) for the two-dimensional Swift-Hohenberg equation in \((1)\) with \(\nu = 1.6\). Sample contour plots of the profiles are provided at folds along the curves. Courtesy of David Lloyd.
Recent work has identified a new approach for localized dihedral patterns that combines radial spatial dynamics with the finite Fourier approximation in \((2)\) [1, 2]. By expanding \(U\) in the truncated Fourier series in \((2)\) and projecting onto each Fourier mode, the two-dimensional (2D) PDE in \((1)\) becomes an (\(N+1\))-dimensional system of radial ODEs to which we can apply radial normal form theory [5, 7, 8]. This approach proves that the finite Fourier approximation of \((1)\) (with truncation order \(N\)) exhibits a localized \(\mathbb{D}_m\) pattern that bifurcates from the trivial state [1], provided that we can find a nontrivial solution to the quadratic algebraic system
\[a_{n} = 2\sum_{j=1}^{N-n} \cos\left(\frac{m\pi(n-j)}{3}\right) a_{j} a_{n+j} + \sum_{j=0}^{n} \cos\left(\frac{m\pi(n-2j)}{3}\right) a_{j}a_{n-j},\qquad n=0,1,\ldots, N. \tag3\]
We can solve this quadratic system explicitly for \(N=1,2,3\), thus revealing a wide range of nontrivial patches that will bifurcate from \(\mu=0\) (see Figure 3). Interestingly, if \(m\) is a multiple of six (including the hexagon case), we can show that localized patches bifurcate for sufficiently large \(N\) [1]. We can also extend these results to obtain dihedral ring patterns that bifurcate from the trivial state at \(\mu=0\) [2]; in this case, the problem reduces to finding a nontrivial solution to a cubic algebraic system.
Figure 3. Localized dihedral patches when \(N=2\) in \((3)\). There are two solutions each for rhombic \((R) \sim \mathbb{D}_{2}\), triangular \((T) \sim \mathbb{D}_{3}\), square \((S) \sim \mathbb{D}_{4}\), and hexagonal \((H) \sim \mathbb{D}_{6}\) patterns. Figure courtesy of [1] under the
Creative Commons Attribution 3.0 license.
Open Problems and the Future
Proving that the finite Fourier approach accurately approximates the 2D SHE is still a major challenge, as is finding a direct proof that localized dihedral patterns bifurcate from the trivial state at \(\mu=0\). It is perhaps more realistic to extend recent results [1, 2] to 3D localized patterns. Meanwhile, finding localized hexagons in the ferrofluid problem remains a tantalizing prospect. As evident in Figure 2, the bifurcation curves of the localized hexagon are significantly more complex than those of the axisymmetric spot, though an explanation of this discrepancy is currently beyond our mathematical prowess. Ultimately, however, this field has made tremendous progress in the last 20 years, due in part to the increasing capabilities of numerical simulations. As we continue to study the emergence and behavior of localized multidimensional patterns, our computational results will raise new and interesting questions for mathematicians.
References
[1] Hill, D.J., Bramburger, J.J., & Lloyd, D.J.B. (2023). Approximate localised dihedral patterns near a Turing instability. Nonlinearity, 36(5), 2567.
[2] Hill, D.J., Bramburger, J.J., & Lloyd, D.J.B. (2024). Dihedral rings of patterns emerging from a Turing bifurcation. Nonlinearity, 37(3), 035015.
[3] Hill, D.J., Lloyd, D.J.B., & Turner, M.R. (2021). Localised radial patterns on the free surface of a ferrofluid. J. Nonlinear Sci., 31(5), 79.
[4] Lioubashevski, O., Hamiel, Y., Agnon, A., Reches, Z., & Fineberg, J. (1999). Oscillons and propagating solitary waves in a vertically vibrated colloidal suspension. Phys. Rev. Lett., 83(16), 3190-3193.
[5] Lloyd, D., & Sandstede, B. (2009). Localized radial solutions of the Swift-Hohenberg equation. Nonlinearity, 22(2), 485.
[6] Lloyd, D.J.B., Sandstede, B., Avitabile, D., & Champneys, A.R. (2008). Localized hexagon patterns of the planar Swift-Hohenberg equation. SIAM J. Appl. Dyn. Syst., 7(3), 1049-1100.
[7] McCalla, S.G., & Sandstede, B. (2013). Spots in the Swift-Hohenberg equation. SIAM J. Appl. Dyn. Syst., 12(2), 831-877.
[8] Scheel, A. (2003). Radially symmetric patterns of reaction-diffusion systems. Mem. Amer. Math. Soc., 165(786), 1-86.
[9] Swift, J., & Hohenberg, P.C. (1977). Hydrodynamic fluctuations at the convective instability. Phys. Rev. A, 15(1), 319-328.
[10] Turing, A.M. (1952). The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B, 237(641), 37-72.
Dan J. Hill is an Alexander von Humboldt Postdoctoral Fellow at Saarland University in Germany. David J.B. Lloyd is a professor of mathematics at the University of Surrey in England. He received the 2024 T. Brooke Benjamin Prize in Nonlinear Waves from SIAM.