Formation of square lattices in coupled pattern-forming systems

Christopher Strickland,Patrick D Shipman,Daniel A Pearson

doi:10.11145/j.biomath.2016.12.181

Abstract

A wide variety of natural and labo-ratory systems can produce patterns of ripples, hexagons, or squares. The formation of stable square patterns from partial differential equation models requires specific cubic nonlinearities involving higher-order derivatives. Motivated by plant phyllotaxis, we demonstrate that the coupling of more than one pattern-forming system can produce square patterns without these special nonlinearities.

Highlights

LATTICE PATTERNS INNATURAL AND LABORATORY SYSTEMSPatterns of ripples (Fig. 1 (a)) or hexagons (Fig. 1 (b)) are observed in a wide variety of natural and laboratory systems
On the other hand, are evident in the surface morphology of the cactus of Fig. 2 (b). Both ripples and hexagons may be observed in RaleighBenard convection experiments [8], the Rosenzweig instability in ferrofluids [12], nanoscale structures formed by bombarding a binary material by a broad ion beam [1], [2], [7], [29], geological formations [17], and landscape-scale vegetation patterns in drylands [4], [6], [18], [19]
We propose in this paper an alternative to cubic terms involving higher-order derivatives, namely the coupling of two pattern-forming systems, that can result in a pattern of squares

Summary

INTRODUCTION

Patterns of ripples (Fig. 1 (a)) or hexagons (Fig. 1 (b)) are observed in a wide variety of natural and laboratory systems. Mathematical analysis of Equation (1) proceeds by first performing a linear stability analysis of the homogeneous steady-state solution u = 0 This determines the modulus kc of wavevectors that will be present in the pattern. The key observation is that the square patterns that are evident at radii r = r1 and r = r5 are formed not by only two Fourier modes with wavevectors as in Fig. 3 (b), but by overlapping triads of modes that satisfy summation relations similar to the wavevectors of Fig. 3 (a) that produce hexagons. These are the wavevectors corresponding to the smaller amplitudes.

A mechanistic model for phyllotactic patterns

SQUARE PATTERN FORMATION IN COUPLED EQUATIONS OF SWIFT-HOHENBERG TYPE