Interaction of spiral waves in the general complex Ginzburg-Landau equation

Abstract

Many physical systems have the property that its dynamics is driven by some kind of spatial diffusion that is in competition with a reaction, like for instance two chemical species that react at the same time that there is a diffusion of each of them into the other, This interplay between reaction and diffusion produce non-homogeneous patterns that can sometimes be very rich. The mathematical models that describe this kind of behaviours are usually nonlinear partial differential equations whose solutions represent these patterns. In this thesis we focus on an specific reaction-diffusion equation that is the so-called general complex Ginzburg-Landau equation that is used as a model for oscillatory systems in extended domains. In particular we are interested in the type of patterns in two dimensions that arise when the solutions have a non-vanishing Brouwer degree. These patterns have the property that they exhibit rotating waves in the shape of spirals, which means that the contour lines arrange in the shape of spirals that emerge from the points where the solution vanishes. When the solution vanishes only at one point all the time dependence appears as a frequency term so the solutions can be expressed as a function of the polar radius and n. .¿Öt where n is the degree of the solution, . is the polar angle and ¿Ö is the frequency of the wave. Therefore, these solutions can be expressed in terms of a system of ordinary differential equations. These solutions do only exist with a given frequency, and as a consequence and due to the existence of a dispersion relation, the wavenumber far from the origin, the so-called asymptotic wavenumber, is also unique. When the solutions have more than one isolated zero, the condition on the degree of the function has the effect of producing several spirals that emerge from the different zeros of the solution. These spirals evolve in time keeping their structure but moving around on the plane. In this work we use asymptotic ana