Discrete Solitons: Methods, Theory and Applications

本建议的目的是使用一些数学工具和技术（哈密顿范式，变分分析，指数渐近，可积性方法，多尺度技术，均匀化和埃文斯函数方法等），结合数值方法(延拓和分岔理论工具，牛顿型方法，结合数值线性稳定性分析和直接时间积分)来系统地探索离散系统中的非线性波。特别是，我们计划在晶格设置中解决波的行为，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。早期的研究大多是在1+1（1个空间，1个时间）维度上进行的。我们现在甚至可以在3+1维度和检查物理现实的设置。(2)无序、非线性和离散之间的相互作用。众所周知，紊乱会导致局部化。了解这种机制与非线性的相互作用，特别是在现实的离散设置中，是至关重要的，因为缺陷在物理系统中无处不在。(3)离散系统中的行波也是最重要的。本征局域模（ILM’s）具有令人感兴趣的能量瓶颈特性。但是它们能把它从一个分子带到另一个分子（从一个晶格点到另一个晶格点）吗？如果是这样，它们将是许多生物能量过程的天然候选者，比如光合作用。(4)这种波的不稳定性研究我们相信，我们现在已经接近于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潜在对称性联系起来。(5)最后，对越来越复杂的物理模型的理解也令人感兴趣。后者涉及额外的物理扰动，例如，远程相互作用，边界条件，它们之间的多波相互作用以及与缺陷的相互作用。在过去的十年里，由于内在局域模式（ILM’s）在能量局域化和输运中的作用已经在各种情况下得到了重视，因此一直是一个越来越受关注的话题。它们的应用范围包括非线性光学和电信（光纤和波导）、原子物理学（BEC，这是2001年诺贝尔物理学奖因其实验观察而突出的一个基本相关问题）、凝聚态物理学（超导和电荷密度波）、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）。感兴趣的领域是多样而广泛的，对基础物理的理解的影响可能非常深刻，但潜在的数学原理是简单而统一的。这些模型具有非线性复杂行为的共同结构，我们希望探讨其时空变化。理解这种行为和（非线性）波在其中的作用是所有这些领域的根本兴趣。非线性波在光学中表示光的电场，在BEC中表示量子力学玻色粒子波函数，在大气中表示DNA碱基对距离或成核液滴的液-气界面。因此，这些波的性质、稳定性、动态演变和内部结构是大量物理效应的核心。我们的研究目标是利用分析和计算数学技术以及物理直觉的结合来探索这些特征。我们的项目特别关注晶格动力系统，其中空间是离散的，就像在许多基本应用中一样，例如光波导、DNA和超导约瑟夫森结阵列。

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.