On three different manifestations of instability of fronts in parabolic and partially parabolic systems

行波是偏微分方程的解，在沿优选方向移动时保持其形状。行波是偏微分方程中的基本相干结构，并且它们经常充当复杂图案的构建块。这个项目是关于抛物和部分抛物系统中的行波（在部分抛物系统中，一些量扩散，另一些不扩散）。更准确地说，本项目的重点是研究行波不稳定性的三种现象学上不同的表现：（1）如果一个系统支持以不同速度移动的波，什么样的不稳定性会导致从一个波到另一个波的转变？在这样的耦合方程组中，优选的波速是由线性动力学还是非线性动力学定义的？(2)如何用分析方法证明或预测不稳定是对流性的？对于对流不稳定波，小扰动离开波界面的速度比它们的增长速度快。(3)不稳定性有时表现为新结构的出现。这些可以是波的渐近静止状态附近的模式，或者是新的全局结构，例如调制波，调制波是由波和周期性模式组成的复合波。已知调制波存在于抛物系统中。该项目的一个组成部分是致力于部分抛物系统中调制波的研究。行波在自然界和人类活动中大量存在。它们出现在不同领域的应用问题中：光通信，燃烧理论，生物数学，化学，种群动力学，仅举几例。关于行波是否存在、系统中有多少行波以及它们在扰动下的弹性的信息有助于理解复杂现象。拟议的工作将有助于确定存在多个波的情况，以及可以预测和控制它们之间的过渡的情况。在真实的生活中观察到的行波通常是稳定的（在小扰动下有弹性）。稳定的波可以承受携带它们的介质的不均匀性和可能由各种因素引起的扰动。研究行波的科学家和工程师有兴趣知道波在什么参数范围内是稳定的，但同样重要的是要知道不稳定波的预期。拟议的工作将允许详细和严格的结论，得出不同的扰动波的行为和不稳定的结果是什么。

Traveling waves are solutions of partial differential equations that preserve their shape while moving in a preferred direction. Traveling waves are basic coherent structures in partial differential equations and they often serve as building blocks for complex patterns. This project is about traveling waves in parabolic and partly parabolic systems (in partly parabolic systems some quantities diffuse and others do not). More precisely, this project is focused on the investigation of three phenomenologically different manifestations of instability of traveling waves: (1) If a system supports waves that move with different speeds, what kind of instability can cause a transition from one wave to another? In such systems of coupled equations, will the preferred wave speed be defined by linear or nonlinear dynamics? (2) How to analytically prove or predict that instability is convective? For a convectively unstable wave small perturbations are moving away from the interface of the wave faster than they grow. (3) An instability sometimes manifests itself in the appearance of new structures. These can be patterns in a neighborhood of an asymptotic rest state of the wave, or new global structures such as modulated waves which are composite waves that consist of a wave and periodic patterns. Modulated waves are known to exist in parabolic systems. A component of this project is devoted to the study of modulated waves in partly parabolic systems. Traveling waves are abundant in nature and human activities. They arise in applied problems from different fields: optical communication, combustion theory, biomathematics, chemistry, population dynamics, to name a few. Information about whether traveling waves exist, how many of them are there in the system, and their resilience under perturbations assists in understanding of complex phenomena. Proposed work will help to identify situations when multiple waves exist and when the transition between them can be anticipated and controlled. Traveling waves that are observed in real life are usually stable (resilient under small perturbations). Stable waves withstand inhomogeneity of the medium that carries them and perturbations that may be caused by a variety of factors. Scientists and engineers who study traveling waves are interested to know in what parameter regimes waves are stable, but it is equally important to know what to expect from an unstable wave. The proposed work will allow detailed and rigorous conclusions to be drawn about how different perturbations to a wave behave and what the outcome of an instability is.