Abstract:

In this course I will design the popular m-armed spiral Ansatz for the complex Ginzburg-Landau equation, and prove the existence of such spiral waves in the circular and spherical geometry. The main literature of the course is my doctoral thesis; please see

http://dynamics.mi.fu-berlin.de/preprints/Dai2017-Dissertation.pdf

The time allocation of the course is the following:

1000 - 1200 (First Session)

Part 1. Ignite Talk (10 min and with slides)

I will exhibit several beautiful spiral patterns, explain two mechanisms that trigger spiral patterns, and introduce three mathematical viewpoints to treat spiral waves.

Afterwards I will use a blackboard only.

Part 2. Setting and Main Result (30 min)

Assuming the m-armed spiral Ansatz, I will establish a functional setting for the resulting spiral wave equation and state my main result: the existence of Ginzburg-Landau spiral waves. Moreover, I will explain the three steps in my proof: global bifurcation analysis, perturbation arguments, and determination of types of pattern.  

Part 3. Design of Spiral Ansatz (60 min)

I will define a tip of a spiral as a phase singularity that can be described by the winding number. Then I design the spiral Ansätze from symmetry perspective. Hence I need to introduce some basic knowledge of equivariant dynamical systems. Viewing an Ansatz as the solution form of a relative equilibrium, I will justify that the m-armed spiral Ansatz is almost the right choice to seek rigidly-rotating Ginzburg-Landau spiral waves.

Part 4. Criterion for Spiral Patterns (20 min)

I will obtain a criterion that determines whether a nontrivial solution of the spiral wave equation exhibits a spiral pattern.

1200 - 1400 (Lunch Time)

1400 - 1700 (Second Session)

Part 5. Global Bifurcation Analysis (90 min)

I will solve the unperturbed case of the spiral wave equation with the following steps. First, I will obtain local bifurcation curves of nontrivial solutions, by showing that the spectrum of the Laplace-Beltrami operator restricted to the subspace of spiral wave solutions consists of simple eigenvalues, only. Second, I will prove the nodal structure of the bifurcation curves. Third, I will derive an a priori bound in sup-norm for nontrivial solutions. Four, I will extend the principal bifurcation curve globally by an open-closed argument.

Part 6. Perturbation Arguments (30 min)

I will introduce the equivariant implicit function theorem, and apply it to prove that the principal bifurcation curve persists for the perturbed cases of the spiral wave equation.

Part 7. Determination of Types of Pattern (40 min)

I will determine the parameter sub-regime that supports spiral waves.

Part 8. Conclusion and Ongoing Research (20 min)

Besides a short conclusion, I will briefly explain a plausible way to prove that other bifurcation curves are also global.