Patterns Arising Through the Continuous Spectrum

9971703 Sandsted非线性波出现在各种应用问题中。一个重要的例子是脉冲和脉冲串在光纤中的传播。其他的例子是波前和螺旋波，例如，在自催化反应或心律失常期间出现。该项目涉及使用分析技术调查这些模式及其相互作用。具体而言，多凸脉冲的存在性和稳定性的偏微分方程的相位对称性，如复杂的Ginzburg-Landau方程解决。另一个问题是要理解小图灵模式和可能导致调制时间周期波的大前沿或脉冲之间的相互作用。虽然脉冲之间的相互作用是服从有限维（虽然不是标准的）微扰分析，它似乎很难解释图灵模式和前线之间的相互作用减少到一个有限的方程组，因为这种相互作用涉及到连续的频谱。连续谱变得重要的另一个问题是平面螺旋波和域边界之间的相互作用。计划更详细地探讨这种相互作用，以便于将平面螺旋波的理论预测与在有界域上进行的实验和数值模拟进行比较。使用光纤进行长距离信息比特传输已成为电信领域的一个重要问题。通过光纤发送的光脉冲对单个比特的信息进行编码。如果光纤能够以稳定的方式支持多脉冲，这些脉冲可以用于编码多比特信息。本计画的一部分是研究脉冲在某些非线性光纤中的传播和相互作用，同时考虑光纤的沿着损耗和放大。

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.