Collaborative research: Nonlinear wave equations and inverse scattering

这个项目涉及一类非线性发展方程(NLEE)，通常被称为孤子方程或可积系统。在过去的五十年里，关于这些系统已经积累了大量的知识，并继续在世界范围内得到广泛的研究。尽管如此，边界条件发挥关键作用的问题仍然构成重大挑战。该项目的目的是在一维和二维空间中对连续和离散的可积NLEE的初边值问题进行广泛的研究，包括它们在几个具体的科学和技术环境中的应用。这项研究将通过发展和使用精确的方法，如逆散射变换和直接方法，结合适当的渐近和数值技术来进行。正在研究的具体问题包括：(I)具有非零边界条件的非线性薛定谔系统(标量和矢量，连续和离散)；(Ii)离散和连续NLEE的边值问题；(Iii)所有上述系统的孤子相互作用的特征；(Iv)(2+1)维可积系统中孤立波结构的分类，如Davey-Stewartson系统。众所周知，非线性波动方程可以模拟从流体动力学和非线性光学到等离子体、宇宙学和量子场论等领域中出现的各种有趣的物理现象。对这些方程的研究特别吸引人，因为它提供了有趣的数学和具体的物理/技术应用的独特组合。特别是，将在这个项目中研究的系统将对波在水波、光纤、激光和玻色-爱因斯坦凝聚体中的传播进行建模。因此，该项目的成果不仅将促进我们对数学的理解，而且还将提供实用信息，帮助科学家和工程师。本科生和研究生的培养也是该项目的一个组成部分。其中一位PIS与人合著了一本关于NLS系统的专著，该专著被广泛引用，并正在帮助传播非线性波领域的知识。这两个PI都有与研究生和本科生合作的长期记录，并将作为该项目的一部分与本科生和博士生合作。

This project concerns a class of nonlinear evolution equations (NLEEs) usually referred to as soliton equations or integrable systems. Over the past fifty years, a large body of knowledge has been accumulated on these systems, which continue to be extensively studied worldwide. Nonetheless, problems in which the boundary conditions play a crucial role still pose significant challenges. This project is aimed at undertaking a wide-ranging investigation of initial-boundary value problems for integrable NLEEs, both continuous and discrete, in both one and two spatial dimensions, including their applications in several concrete scientific and technological settings. This study will be carried out by developing and using exact methods, such as the inverse scattering transform and direct methods, in combination with appropriate asymptotic and numerical techniques. Specific problems that are being studied include: (i) nonlinear Schrodinger systems (scalar and vector, continuous and discrete) with non-zero boundary conditions at space infinity; (ii) boundary value problems for discrete and continuous NLEEs; (iii) characterization of soliton interactions for all of the above systems; (iv) classification of solitary wave structures in (2+1)-dimensional integrable systems such as the Davey-Stewartson system.Nonlinear wave equations are well-known to model a variety of physically interesting phenomena arising in areas ranging from fluid dynamics and nonlinear optics, to plasmas, cosmology and quantum field theory. The study of these equations is especially attractive because it offers a unique combination of interesting mathematics and concrete physical/technological applications. In particular, the systems that will be studied in this project model wave propagation in water waves, optical fibers, lasers and Bose-Einstein condensates. Hence the outcomes of this project will not only advance our mathematical understanding, but they will also provide practical information that will help scientists and engineers. The training of undergraduate and graduate students is also an integral component of this project. One of the PIs has co-authored a monograph on NLS systems that has been referenced extensively and is helping the dissemination of knowledge in the field of nonlinear waves. Both PIs have a long record of working with both graduate and undergraduate students, and will work both with undergraduate students and with Ph.D. students as part of this project.