Formation and Properties of Spatially Localized States

提案ID: 0605238PI: Edgar knoblochininstitution: University of California， berkeley标题：空间定域态的形成与性质摘要这是一项研究空间定域态在一维和二维空间上的形成和稳定性性质的提案。这种状态可以用相空间中的同斜或异斜连接来看待，其中一个空间坐标扮演时间的角色。在许多情况下，这种状态的存在与一种称为同斜形蛇形的现象有关。在本提案中，我们将研究导致空间局部结构形成的不同机制及其相对于一维和二维扰动的稳定性。该研究将采用空间动力学技术对可逆系统与详细的数值相结合，并将重点放在几个四阶偏微分方程在物理科学的重要性。该理论将扩展到可逆性弱破的系统。这一建议侧重于起源和性质的空间局部结构的偏微分方程的重要性，在物理科学。这种结构可以在光学实验、均匀磁场中的铁磁流体实验和对流实验中观察到，并且在均匀的背景中以孤立的凸起形式出现，或者在连接两种不同均匀状态的前沿出现。在振动的颗粒介质和化学反应中可以观察到空间局部振荡，称为振荡子。将发展技术来解释这些状态的起源及其稳定性。该理论将被推广到研究处于轻微倾斜的系统中相关状态的形成。

Proposal ID: 0605238PI: Edgar KnoblochInstitution: University of California, BerkeleyTitle: Formation and Properties of Spatially Localized StatesAbstractThis is a proposal to study the formation and stability properties of spatially localized states in one and two spatial dimensions. Such states can be viewed in terms of homoclinic or heteroclinic connections in phase space, with one spatial coordinate playing the role of time. In many cases the presence of such states is related to a phemenon called homoclinic snaking. In this proposal we shall investigate different mechanisms leading to the formation of spatially localized structures and their stability with respect to both one and two-dimensional perturbations. The study will employ the techniques of spatial dynamics for reversible systems coupled with detailed numerics, and will focus on several fourth order partial differential equations of importance in the physical sciences. The theory will be extended to systems in which reversibility is weakly broken.This proposal focuses on the origin and properties of spatially localized structures in partial differential equations of importance in the physical sciences. Such structures are observed in experiments in optics, ferrofluids in a uniform magnetic field and in convection, and take the form of isolated bumps in an otherwise uniform background, or a front connecting two different uniform states. Spatially localized oscillations, called oscillons, are observed in vibrating granular media and in chemical reactions. Techniques will be developed to explain the origin of these states, and their stability properties. The theory will be extended to study the formation of related states in systems placed on a slight incline.