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PhD Candidate: Iain R. Moyles
Mathematics, UBC
Wed 27 May 2015, 12:30pm SPECIAL
One Time Event
Room 203 of the Graduate Student Centre, 6371 Crescent Rd., UBC
Exam: Hybrid Asymptotic-Numerical Analysis of Pattern Formation Problems
Room 203 of the Graduate Student Centre, 6371 Crescent Rd., UBC
Wed 27 May 2015, 12:30pm-2:30pm


ABSTRACT:  We present an analysis of the Gierer-Meinhardt model with saturation (GMS) on various curve geometries in two-dimensions. We derive a boundary fitted coordinate framework which translates an asymptotic two-component differential equation into a single component reaction diffusion equation with singular interface conditions. We create a numerical method that generalizes the solution of such a system to arbitrary two-dimensional curves and show how it extends to other models with singularity properties that are related to the Laplace operator. This numerical method is based on integrating logarithmic singularities which we handle by the method of product integration where logarithmic singularities are handled analytically with numerically interpolated densities.

In parallel with the numerical method, we present some analytical solutions to the GMS model on circular and slightly perturbed circular curve geometries. We see that for the regular circle, saturation leads to a hysteresis effect for two dynamically stable branches of equilibrium radii. For the near circle, we show that there are two distinct perturbations to the velocity profile, one which introduces angular dependence, and one which introduces a vertical shift caused by quadratic Fourier mode interactions. We perform a linear stability analysis to the true circle solution and show that there are two classes of eigenvalues leading to breakup or zigzag instabilities. For the breakup instabilities we show that the saturation parameter can completely stabilize perturbations that we show are always unstable without saturation and for the zigzag instabilities we show that the eigenvalues are given by the near-circle curve normal velocity. The breakup analysis is based on the reduction of an implicit non-local eigenvalue problem (NLEP) to a root finding problem. We derive conditions for which this eigenvalue problem can be made explicit and use it to analyze a stripe and ring geometry. This formulation allows us to classify certain technical properties of NLEPs such as instability bands and a Hopf bifurcation condition analytically.

The results for breakup and zigzag instabilities are verified with numerical simulations of the full model in both stripe and ring geometries. This includes confirmation of dominant breakup modes and demonstrating the stabilizing effect of saturation.
George Bluman
Thu 28 May 2015, 2:00pm
Symmetries and Differential Equations Seminar
Direct construction of conservation laws and connections between symmetries and CLs. Part I
Thu 28 May 2015, 2:00pm-3:00pm


In this first lecture on the construction of the conservation laws for a DE system, we present the Direct Method.  This leads to the direct construction of the CLs for essentially any DE system in a systematic framework.  The classical Noether's Theorem only works for variational systems and also requires the construction of the Lagrangian.  The Direct Method involves working directly with a given DE system.  The second lecture will show explicitly how the Direct Method generalizes Noether's Theorem and overcomes all of its limitations.