Discrete Solitons: Methods, Theory and Applications

离散孤子：方法、理论和应用

• 批准号: 0204585
• 负责人: Panayotis Kevrekidis
• 金额: $ 12.6万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-01 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204585&HistoricalAwards=false
• 关键词: Discrete; Solitons; Methods; Theory; Applications

The aim of this proposal is to use a number of mathematical tools and techniques (Hamiltonian normal forms, variational analysis, exponential asymptotics, integrability methods, multiscale techniques, homogenization and Evans function methods among others), in conjunction with numerical methods (continuation and bifurcation theory tools, Newton type methods together with numerical linear stability analysis and direct time integration) to systematically explore nonlinear waves in discrete systems. In particular, we plan to address the following aspects of the behavior, dynamics and stability of the waves in lattice settings: (1) The role of spatial dimensionality on nonlinear waves. Most of the earlier studies had been conducted in 1+1 (1 space, 1 time) dimension. We can now go even in 3+1 dimensions and examine physically realistic settings. (2) The interplay between disorder, nonlinearity and discreteness. It is well known that disorder can induce localization. Understanding the interplay of this mechanism with nonlinearity, especially in realistic discrete settings is then crucial, as defects are ubiquitous in physical systems. (3) Travelling waves in discrete systems are also of paramount importance. Intrinsic Localized Modes (ILM's) have the intriguing property of bottlenecking the energy. But could they carry it over (even more so in a targeted way) from one molecule to another (from one lattice site to another)? If so, they would be natural candidates for many bioenergetics processes, such as photosynthesis. (4) The study of instabilities of such waves. We believe that we are now close to a general classification of the possible instabilities and to a connection of these with the underlying symmetries of the physical problem. (5) Finally, the comprehension of progressively more complex physical models is also of interest. The latter involve additional physical perturbations such as, for example, long range interactions, boundary conditions, the interaction of multiple waves between them and with defects. Intrinsic Localized Modes (ILM's) have been a topic of increasing focus over the past decade as their role in energy localization and transport has been appreciated in a variety of contexts. Their applications span nonlinear optics and telecommunications (optical fibers and waveguides), atomic physics (BEC, an issue of fundamental relevance as highlighted by the Physics Nobel Prize in 2001 awarded for its experimental observation), condensed matter physics (superconductivity and charge density waves), biophysics (the local breaking of DNA and conformational changes in proteins), and environmental science (nucleation of liquid droplets in the atmosphere). The areas of interest are diverse and broad, the impact of the understanding of the fundamental physics is potentially very deep, but the underlying mathematical principles are simple and unifying. These models share a common structure of nonlinear complex behavior, the spatio-temporal variation of which we wish to explore. Understanding this behavior and the role of (nonlinear) waves in it is of fundamental interest in all these fields. The nonlinear waves represent the electric field of light in optics, the quantum-mechanical Bose particle wavefunction in BEC, the DNA base-pair distance or the liquid-vapor interface of a nucleating droplet in the atmosphere. The properties of these waves, their stability, dynamical evolution and internal structure are therefore at the heart of a wealth of physical effects. The goal of our research is to explore these features using a combination of analytical and computational mathematical techniques and physical intuition. Our project addresses, in particular, lattice dynamical systems, where space is discrete, as is the case in many fundamental applications, such as optical waveguides, DNA, and arrays of superconducting Josephson junctions.

该提案的目的是结合数值方法（连续和分岔）使用多种数学工具和技术（哈密尔顿范式、变分分析、指数渐进、可积方法、多尺度技术、均质化和埃文斯函数方法等）理论工具、牛顿型方法以及数值线性稳定性分析和直接时间积分）系统地探索离散系统中的非线性波。特别是，我们计划解决晶格设置中波的行为、动力学和稳定性的以下方面：（1）空间维度对非线性波的作用。大多数早期研究都是在1+1（1空间，1时间）维度进行的。 我们现在甚至可以进入 3+1 维度并检查物理真实设置。 (2)无序性、非线性和离散性之间的相互作用。 众所周知，紊乱可以诱导定位。 了解这种机制与非线性的相互作用，尤其是在现实的离散设置中至关重要，因为缺陷在物理系统中无处不在。 (3) 离散系统中的行波也至关重要。本征局域模式 (ILM) 具有令人着迷的能量瓶颈特性。 但他们能否将其从一个分子转移到另一个分子（从一个晶格位点转移到另一个分子位点）（甚至以有针对性的方式）？ 如果是这样，它们将成为许多生物能过程的天然候选者，例如光合作用。 (4)此类波的不稳定性研究。我们相信，我们现在已经接近对可能的不稳定性进行一般分类，并将这些不稳定性与物理问题的基本对称性联系起来。 (5) 最后，理解逐渐复杂的物理模型也很有趣。 后者涉及额外的物理扰动，例如长程相互作用、边界条件、它们之间以及与缺陷的多个波的相互作用。本质局域化模式（ILM）在过去十年中已成为越来越受关注的话题，因为它们在能源局域化和运输中的作用在各种情况下都得到了重视。 它们的应用涵盖非线性光学和电信（光纤和波导）、原子物理学（BEC，2001 年因其实验观察而荣获诺贝尔物理学奖所强调的一个具有根本意义的问题）、凝聚态物理学（超导性和电荷密度波） 、生物物理学（DNA 的局部断裂和蛋白质的构象变化）和环境科学（大气中液滴的成核）。 感兴趣的领域多种多样且广泛，对基础物理学的理解的影响可能非常深刻，但基本的数学原理简单且统一。 这些模型共享非线性复杂行为的共同结构，我们希望探索其时空变化。 了解这种行为以及（非线性）波在其中的作用对于所有这些领域都具有根本意义。 非线性波代表光学中的光电场、BEC 中的量子力学玻色粒子波函数、DNA 碱基对距离或大气中成核液滴的液汽界面。 因此，这些波的特性、稳定性、动力学演化和​​内部结构是大量物理效应的核心。 我们研究的目标是结合分析和计算数学技术以及物理直觉来探索这些特征。 我们的项目特别针对空间离散的晶格动力学系统，就像许多基本应用的情况一样，例如光波导、DNA 和超导约瑟夫森结阵列。