Patterns Arising Through the Continuous Spectrum

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9971703SandstedeNonlinear waves arise in a variety of applied problems. One important example is the propagation of pulses, and pulse trains, in optical fibers. Other examples are fronts and spiral waves that arise, for instance, in autocatalytic reactions or during cardiac arrhythmias. This project is concerned with investigating such patterns and their interactions using analytical techniques. Specifically, the existence and stability of multi-bump pulses in partial differential equations with phase symmetries such as the complex Ginzburg-Landau equation is addressed. Another issue is to understand the interaction between small Turing patterns and large fronts or pulses that may lead to modulated time-periodic waves. While the interaction between pulses is amenable to a finite-dimensional (though not standard) perturbation analysis, it seems difficult to explain the interaction between Turing patterns and fronts by a reduction to a finite set of equations since this interaction involves the continuous spectrum. Another problem where the continuous spectrum becomes important is the interaction between a planar spiral wave and the boundary of a domain. It is planned to explore this interaction in more detail in order to facilitate the comparison of theoretical predictions for planar spiral waves with experiments and numerical simulations that are carried out on bounded domains.Using optical fibers for the transmission of information bits over long distances has become an important issue in telecommunications. An optical pulse that is sent through an optical fiber encodes a single bit of information. If the fiber were capable of supporting multi-bump pulses in a stable fashion, these could be used to encode multiple bits of information. It is part of this project to study the propagation and interaction of pulses in certain nonlinear optical fibers taking loss along the fiber and amplification into account.