Der Ginzburg-Landau Formalismus für musterbildende Systeme mit Erhaltungsgleichungen
Zusammenfassung der Projektergebnisse
We are interested in pattern forming systems with a conservation law exhibiting a Turing instability, a short wave Hopf instability, or a long wave Hopf instability. With the help of a multiple scaling perturbation ansatz a Ginzburg-Landau equation coupled to a scalar conservation law can be derived as an amplitude system for the approximate description of the dynamics of the original system near the first instability. It was the goal of this project to use the amplitude system to establish the global existence of all solutions of the original pattern forming system starting in a small neighborhood of the weakly unstable ground state. We were able to establish such results for reaction-diffusion systems with a conservation law exhibiting a Hopf bifurcation at the wave number k = 0 in case that the system is posed on a large spatial interval with periodic boundary conditions. There was a delay in the project due to the pandemic situation. We strongly expect that similar results can be established in case of a Turing instability or a short wave Hopf instability in the near future. An approach which was not successful for handling the problem without periodic boundary conditions led to two publications, namely attractivity and approximation results for spatially resonant Turing unstable systems, namely with a 2 − 1- and a 3 − 1-resonance. In general the situation on the real line without periodic boundary conditions remained unclear. However, we expect that in situations of weak coupling of the pattern forming system and the conservation law long-time existence results can be established.
Projektbezogene Publikationen (Auswahl)
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Modulation theory for pattern forming systems with a spatial 1:2-resonance. Chaos: An Interdisciplinary Journal of Nonlinear Science, 31(5).
Gauß, Nicole; Schneider, Guido; Ali, Sunny Danish & Zimmermann, Dominik
