Global bifurcation of localised 2D patterns emerging from spatial heterogeneity

TL;DR

The paper addresses the long-standing problem of proving the existence of fully localised 2D patterns emerging from a Turing instability in the presence of spatial heterogeneity. It introduces a compact radially symmetric potential in the Swift–Hohenberg equation to render the linearisation Fredholm, enabling Crandall–Rabinowitz local bifurcation analysis and subsequent analytic global continuation (Buffoni–Toland, Chen–Walsh–Wheeler) to large amplitude. It establishes local bifurcation branches in dihedral symmetry classes that interchange axisymmetric spots and non-axisymmetric dipoles as the heterogeneity width $R$ varies, and characterises their stability and bifurcation type. The global bifurcation framework shows the branches are either unbounded, connect to $\\varepsilon=0$, form closed loops, or lose compactness, with numerical examples illustrating distinct global behaviours. Overall, the work provides a rigorous foundation for fully localised 2D patterns induced by compact heterogeneities and suggests avenues for extension to 3D and experiments.

Abstract

We present a general approach to prove the existence, both locally and globally in amplitude, of fully localised multi-dimensional patterns in partial differential equations containing a compact spatial heterogeneity. While one-dimensional localised patterns induced by spatial heterogeneities have been well-studied, proving the existence of fully localised patterns emerging from a Turing instability in higher dimensions remains a key open problem in pattern formation. In order to demonstrate the approach, we consider the two-dimensional Swift-Hohenberg equation, whose linear bifurcation parameter is perturbed by a radially-symmetric potential function. In this case, the trivial state is unstable in a compact neighbourhood of the origin and linearly stable outside. We prove the existence of local bifurcation branches of fully localised patterns, characterise their stability and bifurcation structure, and then rigorously continue solutions to large amplitude via analytic global bifurcation theory. Notably, the primary bifurcating branch in the Swift-Hohenberg equation alternates between an axisymmetric spot and a non-axisymmetric `dipole' pattern, depending on the width of the spatial heterogeneity.

Figures (8)

• Figure 1: We consider a PDE system appended with a localised potential $V(\mathbf{x})$, causing the otherwise-stable trivial state to destabilise in a compact region and inducing the emergence of localised patterns.
• Figure 2: (a) A schematic bifurcation diagram showing different types of localised dihedral patterns bifurcating off the trivial state. (b) A local bifurcation curve ($b_1$) can be extended to a global bifurcation curve ($b_2$) using analytic bifurcation theory. In particular, the analytic curves that emanate from the local bifurcation either: $(i)$ blow up in their norm, $(ii)$ collide with the boundary $\varepsilon=0$, $(iii)$ form closed loops, or $(iv)$ lose compactness.
• Figure 3: Radial profiles (resp. planar profiles) of linear eigenfunctions $v_{k,n}(r)$ ($u_{k,n}(\mathbf{x})$) are plotted for $R=8$ and (a) $\varepsilon = \varepsilon_{0,1}$, (b) $\varepsilon = \varepsilon_{1,1}$, and (c) $\varepsilon = \varepsilon_{6,1}$.
• Figure 4: (a) The most unstable eigenvalues $\lambda$ of $\mathcal{L}$ for $k=0,1,2,3$ are numerically computed and plotted for $R=5$ and $\varepsilon>0$; (b) - (e) a cartoon of the spectrum of the linear operator $\mathcal{L}$ is presented for critical values of $\varepsilon$, where only the essential spectrum and the four most unstable eigenvalues are shown.
• Figure 5: Implicit curves of $F_k(R,\varepsilon,0)=0$ for (a) $k=0,1,\dots,5$ and (b) $k=0,1,\dots,150$. The label $\mathbb{D}_k$ indicates the smallest dihedral group in which the linear eigenfunction $u = u_{k,n}$ lies, and the dotted black lines indicate the asymptotic curve $\varepsilon = \frac{(2n-1)\pi}{2\,R}$. (c) As $R$ passes between $2.6$ and $2.8$, the primary bifurcation swaps from the $\mathbb{D}_{0}$ pattern to the $\mathbb{D}_1$ pattern. (d) At $R=7.5$ the $\mathbb{D}_6$ pattern is an isolated root of $F_k(R,\varepsilon,0)=0$.

Theorems & Definitions (30)

• Remark 1.1
• Remark 1.2
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Remark 3.1
• Lemma 3.2