In this thesis we establish a functional approach to prove the existence of Ginzburg-Landau spiral waves. Based on systematic considerations, we justify the popular m-armed spiral Ansatz by equivariance and the variational structure of the real Ginzburg-Landau equation. This spiral Ansatz transforms the Ginzburg-Landau equation into an elliptic equation. To solve this elliptic equation by our functional approach, we adopt global bifurcation analysis and the result of existence is essentially a consequence of compactness.
The advantage of our functional approach is threefold. First, it avoids smart, but tricky, estimates used in the shooting method. Second, it works for more general underlying spatial domains, not only in the circular geometry, but also in the spherical geometry. Third, it permits the occurrence of a mixed diffusion process when a complex diffusion parameter is introduced. Thus our result of existence of rigidly-rotating spiral waves greatly generalizes those in the literature. Moreover, we prove the existence of two new patterns: frozen spirals in circular and spherical geometries, and 2-tip spirals in the spherical geometry.
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