arXiv subjects

Ryan Goh

Publications and source records attributed to Ryan Goh.

21 records · Page 2Linked to original sources

arXiv2015

Pattern formation in the wake of triggered pushed fronts

Pattern-forming fronts are often controlled by an external stimulus which progresses through a stable medium at a fixed speed, rendering it unstable in its wake. By controlling the speed of excitation, such stimuli, or "triggers," can mediate pattern forming fronts which freely invade an unstable equilibrium and control which pattern is selected. In this work, we analytically and numerically study when the trigger perturbs an oscillatory pushed free front. In such a situation, the resulting patterned front, which we call a pushed trigger front, exhibits a variety of interesting phenomenon, including snaking, non-monotonic wavenumber selection, and hysteresis. Assuming the existence of a generic oscillatory pushed free front, we use heteroclinic bifurcation techniques to prove the existence of trigger fronts in an abstract setting motivated by the spatial dynamics approach. We then derive a leading order expansion for the selected wavenumber in terms of the trigger speed. Furthermore, we show that such a bifurcation curve is governed by the difference of certain strong-stable and weakly-stable spatial eigenvalues associated with the decay of the free pushed front. We also study prototypical examples of these phenomena in the cubic-quintic complex Ginzburg Landau equation and a modified Cahn-Hilliard equation.

math.DS↗

arXiv2014

Hopf bifurcation from fronts in the Cahn-Hilliard equation

We study Hopf bifurcation from traveling-front solutions in the Cahn-Hilliard equation. The primary front is induced by a moving source term. Models of this form have been used to study a variety of physical phenomena, including pattern formation in chemical deposition and precipitation processes. Technically, we study bifurcation in the presence of essential spectrum. We contribute a simple and direct functional analytic method and determine bifurcation coefficients explicitly. Our approach uses exponential weights to recover Fredholm properties and spectral flow ideas to compute Fredholm indices. Simple mass conservation helps compensate for negative indices. We also construct an explicit, prototypical example, prove the existence of a bifurcating front, and determine the direction of bifurcation.

math.AP↗

arXiv2013

Triggered Fronts in the Complex Ginzburg Landau Equation

We study patterns that arise in the wake of an externally triggered, spatially propagating instability in the complex Ginzburg-Landau equation. We model the trigger by a spatial inhomogeneity moving with constant speed. In the comoving frame, the trivial state is unstable to the left of the trigger and stable to the right. At the trigger location, spatio-temporally periodic wavetrains nucleate. Our results show existence of coherent, "heteroclinic" profiles when the speed of the trigger is slightly below the speed of a free front in the unstable medium. Our results also give expansions for the wavenumber of wavetrains selected by these coherent fronts. A numerical comparison yields very good agreement with observations, even for moderate trigger speeds. Technically, our results provide a heteroclinic bifurcation study involving an equilibrium with an algebraically double pair of complex eigenvalues. We use geometric desingularization and invariant foliations to describe the unfolding. Leading order terms are determined by a condition of oscillations in a projectivized flow, which can be found by intersecting absolute spectra with the imaginary axis.

math.DS↗