Research
Pattern formation
Pattern formation is an ubiquitous natural phenomenon occurring in many areas of natural sciences. Some examples include animal skin patterns, cell differentiation, spiral waves in chemical reactions, propagation of flame fronts, laser interference patterns and sea shell patterns. Often, these patterns have very complex dynamics including moving and oscillating spots, self-replication and coarsening. My research explores the spatio-temporal dynamics of patterns such as spots or interfaces in those models. Analytically, We use a combination of analytic and numerical methods to obtain critical thresholds for various types of instablities. Furthurmore, for stable moving patterns, we reduce the partial differential equation (PDE) model to a set of ordinary diffential equations (ODE) w.r.t the pattern location. Then we can treat the interaction of localized structure as particle interaction, a further approximation is possible for large number of spots by using techniques from collective behaviour in biological and physical systems.
Research highlights
Moving and jumping spot
We consider a single spot solution for the Schnakenberg model in a two-dimensional unit disk in the singularly perturbed limit of a small diffusivity ratio. For large values of the reaction-time constant, this spot can undergo two different types of instabilities, both due to a Hopf bifurcation. The first type induces oscillatory instability in the height of the spot. The second type induces a periodic motion of the spot center. See Fig. 1. We use formal asymptotics to investigate when these instabilities are triggered, and which one dominates. In the parameter regime where spot motion occurs, we construct a periodic solution consisting of a rotating spot, and compute its radius of rotation and angular velocity. see paper "Moving and jumping spot in a two-dimensional reaction–diffusion model " for detail.
Fig 1: The left picture shows the jumping and moving spot. The right picture shows the trajectory of the moving spot and prediction.
Multiple vortices
We consider vortex dynamics in the context of Bose–Einstein condensates (BECs) with a rotating trap, with or without anisotropy. vortex-free steady state become unstable as the rotation is increased above a critical threshold. See the Movie. In the limit of many vortices, BECs are known to form vortex crystal structures, whereby vortices tend to arrange themselves in a hexagonal-like spatial configuration. Using our asymptotic reduction, we derive the effective vortex crystal density and its radius. We also obtain an asymptotic estimate for the maximum number of vortices as a function of rotation rate. see paper "Vortex precession dynamics in general radially symmetric potential traps in two-dimensional atomic Bose-Einstein condensates" for detail.
Fig.3: Comparison from PDE simulation and our approximation
Fig.4: Esimate for maximum number of vortices as a function of rotation rate.
Oscillating Spikes
In an extended Schnankberg model, we find that spike positions can oscillate in a collective way. There are [N/2]+1 types of oscillations when N translation modes are excited. The final state of oscillation only depends on the initial state.
Six spikes have three oscillatory states.
Five spikes have four types of oscillatory states.
Stationary and Moving N-Spot-Ring solutions
We find an N-spot ring solution consisting of N spots with oscillatory tails uniformly distributed on a ring. Those bound states can either start to rotate or travel at a uniform speed when the bifurcation parameter is increased above some threshold.
Oscillating spots in 2-D
Pattern recognition
Pattern recognition has its origins in statistics and engineering, it aims at recognizing pattern and regularities from all sorts of data. My research involving in finding the key charactertics in data and classfiy data sets.
Classification of epoxy resins under different processing methods
The left Figure is X-ray CT images of four samples of epoxy resins cut out from the left cube. Although each sample was produced by the different process, it is difficult to distinguish what is the essential difference among them by our naked eyes. We use SVD (singular vakue decomposition) method to classify the four samples as blew. We interpret the first principle component as the key feature, which corresponds to the total variation of the image and make a connection with its macroscopic material properties. see papaer "Bridging a mesoscopic inhomogeneity to macroscopic performance of amorphous materials in the framework of the phase field modeling" for detail.
