Lecture Room B, 4th Floor, The 3rd General Building, NTHU

(清華大學綜合三館 4樓B演講室)

Spiral Waves in the Circular and Spherical Geometries: the Ginzburg-Landau Paradigm

Jia-Yuan Dai (Freie Universität Berlin &
NCTS)

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.