Discrete Solitons: Methods, Theory and Applications

这一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、齐次化和Evans函数方法等)，结合数值方法(延拓和分叉理论工具、牛顿型方法和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动。特别是，我们计划研究格子环境中波的行为、动力学和稳定性的以下几个方面：(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空间，1时间)维度进行的。我们现在甚至可以在3+1维空间中检查物理上真实的设置。(2)无序性、非线性和离散性的相互作用。众所周知，无序可以导致局部化。因此，理解这种机制与非线性的相互作用，特别是在现实的离散环境中，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非常重要的。本征局域模(ILM‘s)具有阻碍能量瓶颈的有趣特性。但他们能把它从一个分子带到另一个分子(从一个格子位置到另一个格子位置)吗？如果是这样的话，它们将是许多生物能量学过程的天然候选者，例如光合作用。(4)这类波的不稳定性研究。我们相信，我们现在已经接近了对可能的不稳定性的一般分类，以及这些不稳定与物理问题的基本对称性之间的联系。(5)最后，对逐渐复杂的物理模型的理解也是有意义的。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之间的多个波的相互作用。在过去的十年中，固有局域化模式(ILM‘s)在能源局部化和传输中的作用受到越来越多的关注，因为它们在各种情况下都得到了重视。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(BEC，2001年因其实验观测而获得的诺贝尔物理学奖强调的一个基本相关问题)、凝聚态物理(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)以及环境科学(大气中液滴的成核)。感兴趣的领域是多样和广泛的，对基础物理学的理解的影响可能非常深刻，但基本的数学原理是简单和统一的。这些模型共享一个非线性复杂行为的共同结构，我们希望探索其时空变化。了解这种行为和(非线性)波在其中的作用是所有这些领域的基本兴趣。非线性波代表光学中的光场、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气液界面。因此，这些波的性质、稳定性、动力演化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组合来探索这些特征。我们的项目特别针对晶格动力学系统，其中空间是离散的，就像在许多基本应用中的情况一样，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.