Asymptotic Methods for Singularly Perturbed Nonlinear Systems

本文主要包括两大主题：A)聚焦非线性薛定谔方程（NLS）的半经典（小色散）极限及其相关问题；B)系统在奇异扰动下的可积动力学的持续性，包括同斜/异斜解。A) Bronski和McLauchlin的数值实验表明，在聚焦NLS的小色散极限处形成了一个剧烈的无组织振荡区域。利用黎曼-希尔伯特问题（RHP）的方法，我们找到了一种方法，通过与问题相关的超椭圆曲面的演变，跟踪我们的初始数据到“剧烈和无组织振荡”区域的演变。该表面的格值变化对应于初始数据演化过程中的相变。在与Venakides和Zhou的联合工作中，对纯辐射情况进行了全面的研究。在这里，我们建议研究相变在最困难的情况下，包括孤子和辐射。B)最近（Tovbis, Pelinovsky）证明了奇摄动第5阶Kortveg - de Vries （KdV）方程的同斜/异斜解的持久性可以用某（阶）重标系统的Stokes常数表示。提出的目标是为其他类型的解决方案和系统开发相应的技术，包括NLS和其他模型的格上的静止和移动行波。众所周知，只有一小部分用于模拟现实世界问题的非线性系统允许显式形式的数学解决方案，留下近似方法和计算机模拟成为最常用的工具。渐近方法在近似方法中扮演着重要的角色，因为它通常更容易在某些渐近极限（例如，无限时间）下研究系统，然后允许对大但有限的时间值进行一些“小”修正。这种方法被使用了几个世纪，例如，在天神力学中。然而，如果原始系统和极限系统表现出性质不同的行为，“小”修正不一定是小的。例如，二阶非线性振子的弱耦合系统一般具有混沌行为，而非耦合振子的极限系统不具有混沌行为。这类系统称为奇摄动系统，是渐近分析中最困难的问题。目前的建议集中在两种仍在发展中的奇异摄动非线性问题的方法：RHP方法和全阶渐近方法。第一种方法将用于寻找聚焦NLS的小色散极限，这是一个多年未解决的问题。NLS是物理、工程等领域中最常用的非线性波现象模型。第二种方法将用于找出KdV， NLS和一些其他可积模型的行波解在奇异摄动（包括离散化）下是否存在。晶格上的孤子和行波（离散模型）正迅速成为一个非常重要的课题，例如在光纤（包括周期光学结构）、计算数值方法的设计等方面。

The present proposal consists of two major topics: A) semiclassical (small dispersion) limit of the focusing Nonlinear Schroedinger Equation (NLS) and related problems, and; B) persistence of integrable dynamics, including homoclinic/heteroclinic solutions, of a system undergoing a singular perturbation. A) Numerical experiments of Bronski and McLauchlin revealed the formation of a region of violent and disorganized oscillations in the small dispersion limit of the focusing NLS. Using the method of Riemann-Hilbert Problem (RHP), we found a way to track the evolution of our initial data into the region of ``violent and disorganized oscillations" through the evolution of a hyperelliptic surface associated with the problem. Changes of the genus of this surface correspond to phase transitions in the evolution of our initial data. The pure radiational case was studied completely in the joint work with Venakides and Zhou. Here we propose to study phase transitions in the most difficult case that includes both solitons and radiation. B) It has been recently proved (Tovbis, Pelinovsky) that persistence of homoclinic/heteroclinic solutions to singularly perturbed 5th Kortveg - de Vries (KdV) equation can be expressed in terms of the Stokes' constants of certain (leading order) rescaled system. The proposed goal is to develop the corresponding technique to other types of solutions and systems, including stationary and moving travelling waves on lattices for NLS and other models. It is well known that only a tiny portion of nonlinear systems used to model real world problems allows for explicit form mathematical solutions, leaving approximate methods and computer simulations to be the most used tools. Asymptotic methods play a prominent role among approximate methods, as it is often easier to study a system in some asymptotic limit (say, infinite time) and then to allow some ``small" corrections for large but finite values of time. This approach was used for centures, for example, in celestian mechanics. However, ``small" corrections are not necessarily small if the original and the limiting systems exibit qualitatively different behavior. For example, a weakly coupled system of 2nd order nonlinear oscillators generically has a chaotic behavior whereas the limiting system of uncoupled oscillators has no chaos. Such systems, called singularly perturbed, are the most difficult subject in the asymptotic analysis. The current proposal is focused on two still developing methods for singularly perturbed nonlinear problems: the method of RHP and of asymptotics beyond all orders. The first method will be used to find small dispersion limit of the focusing NLS - the problem that was open for many years. The NLS is the most frequently used model for nonlinear waves phenomena in physics, engineering, etc. The second method will be used to find when travelling waves solutions to KdV, NLS and some other integrable models survive singular perturbations, including discretizations. Solitons and travelling waves on lattices (discretized models) are rapidly becoming a very important topic, for example, in fiber optics (including periodic optical structures), design of computational numerical methods, etc.