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Localized Patterns

Jason J. Bramburger, Dan J. Hill, David J. B. Lloyd

TL;DR

This survey consolidates the mathematical understanding of localized patterns across dimensions, tracing how finite-wavenumber/Turing instabilities spawn one- and multi-dimensional localized states and detailing the principal mechanisms behind their bifurcation structure, notably homoclinic snaking and front pinning. It foregrounds three core tools—spatial dynamics, exponential asymptotics, and numerical continuation—while illustrating how axisymmetric and dihedral symmetries shape pattern formation, particularly in the Swift–Hohenberg equation and related reaction–diffusion systems. The work emphasizes the gradient/Hamiltonian structure, Maxwell points, and front interactions that govern localization, and it highlights open problems in higher dimensions, nonlocal models, and fully 3D localized states. Overall, the article demonstrates how 1D localization is well-developed, how 2D patches are rich yet still developing, and how 3D localization remains a frontier with substantial theoretical and computational challenges and potential applications in fluids, optics, and materials science.

Abstract

Localized patterns are coherent structures embedded in a quiescent state and occur in both discrete and continuous media across a wide range of applications. While it is well-understood how domain covering patterns (for example stripes and hexagons) emerge from a pattern-forming/Turing instability, analyzing the emergence of their localized counterparts remains a significant challenge. There has been considerable progress in studying localized patterns over the past few decades, often by employing innovative mathematical tools and techniques. In particular, the study of localized pattern formation has benefited greatly from numerical techniques; the continuing advancement in computational power has helped to both identify new types patterns and further our understanding of their behavior. We review recent advances regarding the complex behavior of localized patterns and the mathematical tools that have been developed to understand them, covering various topics from spatial dynamics, exponential asymptotics, and numerical methods. We observe that the mathematical understanding of localized patterns decreases as the spatial dimension increases, thus providing significant open problems that will form the basis for future investigations.

Localized Patterns

TL;DR

This survey consolidates the mathematical understanding of localized patterns across dimensions, tracing how finite-wavenumber/Turing instabilities spawn one- and multi-dimensional localized states and detailing the principal mechanisms behind their bifurcation structure, notably homoclinic snaking and front pinning. It foregrounds three core tools—spatial dynamics, exponential asymptotics, and numerical continuation—while illustrating how axisymmetric and dihedral symmetries shape pattern formation, particularly in the Swift–Hohenberg equation and related reaction–diffusion systems. The work emphasizes the gradient/Hamiltonian structure, Maxwell points, and front interactions that govern localization, and it highlights open problems in higher dimensions, nonlocal models, and fully 3D localized states. Overall, the article demonstrates how 1D localization is well-developed, how 2D patches are rich yet still developing, and how 3D localization remains a frontier with substantial theoretical and computational challenges and potential applications in fluids, optics, and materials science.

Abstract

Localized patterns are coherent structures embedded in a quiescent state and occur in both discrete and continuous media across a wide range of applications. While it is well-understood how domain covering patterns (for example stripes and hexagons) emerge from a pattern-forming/Turing instability, analyzing the emergence of their localized counterparts remains a significant challenge. There has been considerable progress in studying localized patterns over the past few decades, often by employing innovative mathematical tools and techniques. In particular, the study of localized pattern formation has benefited greatly from numerical techniques; the continuing advancement in computational power has helped to both identify new types patterns and further our understanding of their behavior. We review recent advances regarding the complex behavior of localized patterns and the mathematical tools that have been developed to understand them, covering various topics from spatial dynamics, exponential asymptotics, and numerical methods. We observe that the mathematical understanding of localized patterns decreases as the spatial dimension increases, thus providing significant open problems that will form the basis for future investigations.
Paper Structure (35 sections, 6 theorems, 99 equations, 51 figures)

This paper contains 35 sections, 6 theorems, 99 equations, 51 figures.

Table of Contents

  1. Introduction
  2. 1D Localized Pattern Formation
  3. Turing and Finite-Wavenumber Instabilities
  4. Turing's Contribution
  5. General Pattern-Forming Bifurcations
  6. Pattern Formation in the Swift--Hohenberg Equation
  7. Homoclinic snaking
  8. Weakly Nonlinear Analysis
  9. Snakes, Ladders, and Isolas of Localized Patterns
  10. Lattice Dynamical Systems
  11. Variations on the Standard Set-Up
  12. General Reaction-Diffusion Systems
  13. Nonlocal Systems
  14. Slanted Snaking and Finite Domains
  15. Oscillons and Breathers
  16. ...and 20 more sections

Key Result

Theorem 4.1

Fix $\nu\neq0$. Then, there exist constants $\mu_0,r_0,r_1 > 0$ such that SHE_nDim with $n = 2$ has a steady-state solution of the form $U(r) = u_A(r)$, where for each $\mu\in(0,\mu_{0})$, and $J_{0}$ is the $0^\text{th}$ order Bessel function of the first kind.

Figures (51)

  • Figure 1: A collage of localized patterns found in (1) ferrofluids lloyd2015homoclinic, (2) buckling of cylinders (Photo credit: M. A. Wadee), (3) vertically vibrated granular material (Photo credit: P.B. Umbanhowar), (4) Sun spots (By NASA Goddard Space Flight Center from Greenbelt, MD, USA - Solar Archipelago) (5) vegetation patches in arid areas (Photo credit: S. Getzin), and (6) vertically vibrated fluids Lioubashevski1999.
  • Figure 2: Numerically continued bifurcation curves of localized spot (blue) and hexagon patches (yellow) in the 2D SHE \ref{['SwiftHohenberg']} with $\nu = 1.6$. Sample contour plots of the profiles are provided at folds along the curves.
  • Figure 3: Snakes and ladders bifurcation diagrams are a familiar feature of localized patterns in one spatial dimension. The diagram here is for the 1D SHE \ref{['SwiftHohenberg']} with $\nu = 1.6$ and features two intertwined branches of localized solutions: one with a maximum in the middle (blue) and one with a minimum (red). These branches are connected by the asymmetric 'ladder' states (green). As one ascends the diagram, the localized pattern grows by adding new spatially periodic patterns (SPPs) on both sides, as exemplified with the profiles taken at successive saddle-node bifurcations on the curve.
  • Figure 4: Turing patterns are one of the fundamental features of a system that leads to localized patterns. (a) A sketch of solutions just before (left) and after (right) bifurcation (b) A plot of the linear growth curve just before, during, and after the Turing bifurcation.
  • Figure 5: Phase portraits for time-independent solutions of the cubic Ginzburg--Landau equation \ref{['e:GZeqn']} in the (a) subcritical case $b<0$, $\hat{\mu}>0$, and the (b) supercritical case $b>0$, $\hat{\mu}<0$. The system possesses three equilibria $A=0,\pm\sqrt{-\hat{\mu}/b}$ (red circles), and either homoclinic or heteroclinic orbits. We present profiles for (c) localized pulse (blue), (d) front (green), and (e) back (purple) solutions, where the oscillating pattern $U$ (yellow) is bounded by the amplitude $A$
  • ...and 46 more figures

Theorems & Definitions (7)

  • Remark 2.1
  • Theorem 4.1: Spot A, lloyd2009localized
  • Theorem 4.2: Rings, lloyd2009localized
  • Theorem 4.3: Spot B, mccalla2013spots
  • Theorem 4.4: Radial oscillons, mcquighan2014oscillons
  • Theorem 5.1: hill2021localised
  • Theorem 5.2: hill2022dihedral