Computing Interacting Multi-fronts in One Dimensional Real Ginzburg Landau Equations

Abstract

We develop an efficient and robust numerical scheme to compute multi-fronts in one-dimensional Real Ginzburg-Landau equations that range from well-separated to strongly interacting and colliding. The scheme is based on the global centre-manifold reduction where one considers an initial sum of fronts plus a remainder function (not necessarily small) and applying a suitable projection based on the neutral Eigenmodes of each front. Such a scheme efficiently captures the weakly interacting tails of the fronts. Furthermore, as the fronts become strongly interacting, we show how they may be added to the remainder function to accurately compute through collisions. We then present results of our numerical scheme applied to various real Ginzburg Landau equations where we observe colliding fronts, travelling fronts and fronts converging to bound states. Finally, we discuss how this numerical scheme can be extended to general PDE systems and other multi-localised structures.

Weak and strong interaction of excitation kinks in scalar parabolic equations

“…Moreover, as a model for the dynamics far from collisions, we show that under periodic boundary conditions the distances of neighboring kinks asymptotically equalise, and we numerically corroborate that this loss of information happens more broadly. On the other hand, as a quantitative analysis, we derive the ODE in the weak interaction regime following [Ei02,RLZ15] for finitely many kinks and analyse these in some detail, which extends the aforementioned qualitative terrace results. Our analysis relies on blow-up type singular rescaling and identifies the dynamics as being slaved to dynamics on a sphere at infinity, which, for instance, shows that distances become ordered in finite time.…”

“…[CP89,FH89,FM77]. This has been explored in various directions, notably to infinitely many metastable pulses in arbitrary dimension [ZM09]; we mainly follow [Ei02] and [RLZ15]. This allows to derive such ODE rigorously only for sequences of either kinks or antikinks, i.e., monotone initial data, and we therefore restrict attention to (5) with m = 0.…”

“…This appendix is devoted to the derivation of the differential system (17). As explained in the introduction of §3.2, we follow the scheme presented in [Ei02,RLZ15] that we adapt to our situation for the convenience of the reader.…”