Collaborative research RUI: Integrable systems, inverse scattering and applications

这个项目涉及某些普遍存在的非线性波现象的理论和应用研究。 更具体地说，将被研究的系统属于一类称为孤子方程或可积系统的非线性演化方程，它们作为数学模型出现在各种应用领域，从流体动力学和非线性光学，到低温物理学和玻色-爱因斯坦凝聚，仅举几例。 特别是，将研究的模型方程之一描述了在公海中的流氓波的形成。 这些极端事件的特点是波峰高达平均值的四倍，已知会对船只和其他设备造成重大损害。 最近在光纤中也观察到类似于异常波的效应。 这些模型方程的精确数学描述也是光纤传输系统设计中的关键组成部分。 该项目的成果将有助于更好地表征这些系统的行为及其解决方案的性质，它们将阐明这些解决方案在产生流氓波中所起的作用，并将有助于为更深入地理解这些重要的非线性现象铺平道路。该项目将通过研究为布法罗的纽约州立大学的研究生和科罗拉多斯普林斯的科罗拉多大学的本科生提供丰富的教育经验。在数学上，该项目的首要目标是对可积非线性发展方程的初始和初边值问题进行广泛的研究，从而有助于缩小我们对线性和非线性可积系统的知识之间的差距。更确切地说，该项目包括边界条件起关键作用的问题，以及各种奇异极限。将研究的具体问题包括：（i）无穷远处具有非零边界条件的非线性Schroedinger（NLS）型耦合方程组的逆散射变换的发展;（ii）有限区间上具有线性化、周期或近周期边界条件的NLS方程的初边值问题的研究;（iii）研究受调制不稳定性影响的NLS系统的长时间渐近性以及NLS方程和Korteweg-deVries方程在周期边界条件下的半经典极限和无色散极限。

This project involves theoretical and applied investigations of certain ubiquitous nonlinear wave phenomena. More specifically, the systems that will be studied belong to a class of nonlinear evolution equations referred to as soliton equations or integrable systems, which arise as mathematical models in various areas of applications, ranging from fluid dynamics and nonlinear optics, to low-temperature physics and Bose-Einstein condensates, to name a few. In particular, one of the model equations that will be studied describes the formation of rogue waves in the open ocean. These extreme events, which are characterized by wave crests up to four times bigger than the average, have been known to cause significant damages to vessels and other equipment. Effects similar to rogue waves have also been recently observed in optical fibers. A precise mathematical description of these model equations is also a key component in the design of optical fiber transmission systems. The project outcomes will help to better characterize the behavior of these systems and the properties of their solutions, they will elucidate the role that such solutions play in the generation of rogue waves, and they will contribute to paving the way for a deeper understanding of these important nonlinear phenomena. The project will provide a rich educational experience through research for graduate students at the State University of New York at Buffalo and undergraduate students at the University of Colorado at Colorado Springs.Mathematically, the overarching goal of the project is a wide-ranging investigation of initial and initial-boundary value problems for integrable nonlinear evolution equations, thereby contributing to closing the gap between our knowledge of linear and nonlinear integrable systems. More precisely, the project comprises problems where the boundary conditions play a key role, as well as various kinds of singular limits. Specific problems that will be studied include: (i) the development of the inverse scattering transform for coupled systems of equations of nonlinear Schroedinger (NLS) type with non-zero boundary conditions at infinity; (ii) the study of initial-boundary value problems for NLS equations on a finite interval with linearizable, periodic or nearly periodic boundary conditions; (iii) the study of the long-time asymptotics of NLS systems affected by modulational instability and of semiclassical and dispersionless limits for NLS equations and the Korteweg-deVries equation with periodic boundary conditions.