# Pattern Formation

### 1. Formation of Soft-Matter Quasicrystals

Soft matter crystals can be analysed in terms of the interaction potential between two soft-core particles. Such interacting particles form a variety of patterns in the bulk. In addition to forming regular crystalline arrangements, they are also observed to form quasicrystalline patterns – patterns that have rotational symmetry but never repeat in any direction. We investigate the formation and stability of icosahedral quasicrystals using a dynamic phase field crystal model. We identify that appropriate ratio of two length scales and strong resonating nonlinear interactions between them are the required conditions to make quasicrystalline patterns.

**Collaborators:**

– **Morgan Walters**

**Relevant Publications:**

10. Spatially Localized Quasicrystalline Structures, *New Journal of Physics*, 2018.

9. Structural crossover in a model fluid exhibiting two length scales: repercussions for quasicrystal formation, *Physical Review E, 98, 012606, 2018.*

8. Three-dimensional Phase Field Quasicrystals, *Physical Review Letters*, **117**, 075501, 2016.

**Public Outreach:**

– Lunch time talk on ‘*Quasicrystals: minimal recipes and a brief history’* on 11th December 2018.

– Demonstrator for never-repeating patterns at the Big Draw 2016 Festival at the Royal Society, London on October 22nd, 2016.

– Talk at the Leeds Cafe Scientifique on 13th December 2016.

### 2. Localization & Pattern Formation in Fluid Flows

Turbulence transition in shear flows is initiated by patches of turbulence that invade the surrounding laminar flow. Spatially localized exact invariant solutions for different shear flow systems can be obtained efficiently using numerical continuation methods. Dynamically, the occurrence of such spatially localized states can be explained by homoclinic snaking behavior observed in Swift-Hohenberg equation, plane Couette flow, etc.

Emergence of spatio-temporal patterns in systems with linear instability can be effectively analyzed with tools such as amplitude equations. We aim to exploit this feature by selecting the inclined layer convection (ILC) system for analysis. A rich variety of patterns are observed in an ILC system where there is a smooth change from a linearly unstable (buoyancy-driven instabilities) to a linearly stable (shear-driven instabilities) system with variation of the inclination angle. Full DNS simulations of Oberbeck-Boussinesq equations allow us to quantitatively compare different patterns between our numerical simulations and experimental results (see Publication 7). Exact solutions obtained from simulations for small inclinations can then be numerically continued smoothly into shear-dominated regimes (see publication 15).

**Collaborators: **

**Relevant Publications:**

15. Invariant states in inclined layer convection. Part 2. Bifurcations and connections between branches of invariant states, accepted for publication with the *Journal of Fluid Mechanics*, 2020.

7. Spatiotemporal patterns in inclined layer convection, *Journal of Fluid Mechanics*, **794**, 719-745, 2016.

**Public Outreach:**

– Talk at the Superposition on 13th April 2016 with title ‘Patterns in Inclined Layer Convection’

### 3. Role of Mode Interactions in creating complex patterns

Pattern formation in systems in many real world systems such as neural-field models, reaction-diffusion systems and fluid systems such as the Faraday wave system have separation of scales leading to nonlinear modal interactions. A general analysis of possible terms that can arise via modal interactions is subject to both the choice of a lattice grid and the ratio between the two length scales $q$. Motivated by the observance of different grid states and superlattice states in experiments of the Faraday wave system, here we consider a hexagonal lattice grid and identify families of amplitude equations for different values of the ratio in the range $0<q<1/2$. We find that the ratio of $q=1/\sqrt{7}$ gives rise to the maximum number of terms in the amplitude equations (up to third order terms) and that other families of amplitude equations can be recovered by setting the coefficients of certain modal interactions in this `general’ to vanish. For the case with $q=1/\sqrt{7}$, we use equivariant bifurcation analysis to investigate the existence and stability of different patterns over a range of marginally stable growth rates for both the length scales. We also contrast our results with a codimension-1 analysis which assumes that only one of the length scales is marginally stable.

**Relevant Publications:**

16. Mode Interactions and Superlattice Patterns, to be submitted to SIAM Journal of Dynamical Systems, 2020.

**Public Outreach:**

– Talk at PMMH-ESPCI, Paris, April 2019

– Poster at SIAM Conference on Dynamical Systems, May 2019