Asymptotic Methods for Singularly Perturbed Nonlinear Systems

本论文的主要内容包括两个方面：A）聚焦非线性薛定谔方程（NLS）的半经典（小色散）极限及相关问题; B）可积动力学的持久性，包括奇异摄动系统的同宿/异宿解。A）Bronski和McLauchlin的数值实验揭示了在聚焦NLS的小色散极限中形成了一个剧烈和无序振荡的区域。使用黎曼-希尔伯特问题（RHP）的方法，我们找到了一种方法来跟踪我们的初始数据的演变到“暴力和混乱的振荡”的区域，通过与问题相关的超椭圆表面的演变。该表面的变化对应于我们的初始数据的演变中的相变。在与Venakides和Zhou的联合工作中，完全研究了纯辐射情况。在这里，我们建议研究最困难情况下的相变，其中包括孤立子和辐射。B）最近证明了（Tovbis，Pelinovsky）奇摄动五阶KdV方程的同宿/异宿解的持久性可以用某个（首阶）重标度系统的Stokes常数来表示。提出的目标是发展相应的技术，以其他类型的解决方案和系统，包括静态和移动行波的格子上的NLS和其他模型。众所周知，只有一小部分用于模拟真实的世界问题的非线性系统允许显式形式的数学解，留下近似方法和计算机模拟是最常用的工具。渐近方法在近似方法中起着突出的作用，因为它通常更容易在某个渐近极限（例如，无限时间）中研究一个系统，然后允许对大但有限的时间值进行一些"小”修正。这种方法被用于centures，例如，在天体力学。然而，“小”的修正不一定是小的，如果原来的和限制系统exibit定性不同的行为。例如，弱耦合的二阶非线性振子系统一般具有混沌行为，而非耦合振子的极限系统则没有混沌行为。这类系统称为奇摄动系统，是渐近分析中最困难的问题。目前的建议是集中在两个仍在发展中的奇摄动非线性问题的方法：RHP的方法和超越所有订单的渐近。第一种方法将被用来找到小的分散限制的聚焦NLS -这是开放多年的问题。NLS是最常用的模型，在物理，工程等非线性波现象的第二种方法将被用来找到行波解KdV，NLS和其他一些可积模型生存奇异摄动，包括离散化。格子（离散模型）上的孤子和行波正迅速成为一个非常重要的课题，例如在光纤（包括周期性光学结构）、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.