Novel Challenges in Nonlinear Waves and Integrable Systems

本研究涉及非线性波动现象，特别强调研究一类通常被称为孤子方程或可积系统的非线性演化方程。这些系统模拟了不同领域的物理现象，如深水波，等离子体物理，非线性光纤，低温物理和玻色-爱因斯坦凝聚，以及静磁自旋波。所研究的一些方程属于非线性Schrödinger （NLS）方程及其各种推广，它们构成了弱色散非线性波列的通用模型。另一类是描述超短光脉冲在非线性介质中的传播的复短脉冲方程及其推广。这些应用于激光材料加工、激光显微镜、光学时钟和测量、医学（例如，眼科手术和视力矫正）和电信。这些系统的一个显著特征是，除了标准孤子（保持其形状和速度的局部行波）之外，它们还承认所谓的环孤子（不是单值孤子）以及在两者之间振荡的解。本研究旨在阐明的另一个现象是异常波的形成。这些极端事件是异常巨大的表面波，波峰高达平均水平的四倍，不知从哪里出现，消失得无影无踪，因此可能对船舶和海上钻井装置造成重大损害。本研究解决了孤子和调制不稳定性的作用，作为形成异常波的可能的并发机制。最近在光纤实验中也观察到异常波。对布法罗大学研究生的培训以及旨在促进妇女在应用数学领域发展的外联活动将是这个项目的组成部分。在更技术性的层面上，研究了边界条件起关键作用的某些可积系统，以及各种奇异极限。具体任务包括：(i)开发具有无穷远处非零边界条件的各种NLS系统的逆散射变换，特别是与反传播波对应的边界条件；（ii）对玻色-爱因斯坦凝聚体（如孤子串、束缚态、缠绕）和非线性光学（向列液晶等）应用中的具体问题的研究；（iii）严格研究NLS系统的离散谱和谱奇点；（iv）建立了复耦合短脉冲方程的逆散射变换；(v)研究了上述系统的孤子、孤子相互作用和异常波，并将其应用于接近可积系统的非可积系统；（六）网络相变作为孤子方程解的临界行为的研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research concerns nonlinear wave phenomena, with particular emphasis on the study of a class of nonlinear evolution equations commonly referred to as soliton equations or integrable systems. These systems model physical phenomena in diverse fields such as deep-water waves, plasma physics, nonlinear optical fibers, low temperature physics and Bose-Einstein condensates, and magneto-static spin waves. Some of the equations studied belong to the class of the nonlinear Schrödinger (NLS) equation and its various generalizations, which constitute universal models for weakly dispersive nonlinear wave trains. Others belong to the complex short-pulse equation and its generalizations, which describe the propagation in nonlinear media of ultra-short optical pulses. These have applications to laser material processing, laser microscopy, optical clocks and measurements, medicine (e.g., eye surgery and vision correction), and telecommunication. A distinguishing feature of these systems is that, in addition to standard solitons, localized traveling waves that preserve their shape and velocity, they also admit so-called loop solitons, which are not single-valued solitons, as well as solutions that oscillate between the two. Another phenomenon that this research aims at elucidating is the formation of rogue waves. These extreme events are unusually large surface waves with wave crests up to four times the mean level that appear from nowhere and disappear without a trace and can therefore cause significant damage to ships, and offshore drilling units. This research addresses the role of solitons and modulational instability as possible concurrent mechanisms for the formation of rogue waves. Rogue waves have also been recently observed experimentally in optical fibers. The training of graduate students at the University at Buffalo, and outreach activities aiming at promoting the growth of women in applied mathematics will be integral components of this project. At a more technical level, the investigation deals with certain integrable systems in situations in which the boundary conditions play a key role, as well as various kinds of singular limits. Specific tasks include: (i) the development of the inverse scattering transform for various NLS systems with non-zero boundary conditions at infinity, and in particular with boundary conditions corresponding to counter-propagating waves; (ii) the investigation of concrete questions from applications in Bose-Einstein condensates (e.g., trains of solitons, bound states, windings) and in nonlinear optics (nematic liquid crystals, etc); (iii) the rigorous study of the discrete spectrum and spectral singularities of NLS systems; (iv) development of the inverse scattering transform for the complex coupled short-pulse equation; (v) the investigation of solitons, soliton interactions and rogue waves for the above systems, and applications to non-integrable systems in regimes close to the integrable ones; and (vi) the study of phase transitions in networks as critical behavior of solutions of soliton equations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.