Inverse scattering transform outside of classical conditions

研究人员将研究应用中使用的非线性偏微分方程，以模拟流体力学，电信，大气科学，非线性光学，等离子体和天体物理学中的现象。具体的重点是所谓的完全可积系统，应用数学领域也被称为孤子理论。目前已知的大部分都是关于快速衰减或周期性初始条件的，但物理学和应用需要更一般的非标准情况：该项目的重点是了解较慢衰减（或无衰减）初始数据的影响。该项目将提高对流氓波、可积湍流、相干结构在噪声介质中的传播、潮汐波和气象现象（即，牵牛花）。该项目还将提供本科生和研究生水平的研究和指导机会。招聘工作将注意扩大数学科学的研究参与，以及目标学生打算成为高中数学教师。该项目在Korteweg-de弗里斯方程的范围内研究在空间正无穷远处具有较慢衰减（或无衰减）的初始数据对完全可积系统的解的影响，即可以通过逆散射变换（IST）求解和分析的偏微分方程。在正无穷远处慢衰变的情况下，对IST的应用提出了严峻的数学挑战，它的研究是一个未解决的重大开放问题。预测解的长期行为是困难的，因为众所周知的强大的黎曼-希尔伯特机制在这样的初始轮廓上崩溃。该项目的主要工作将致力于扩展黎曼-希尔伯特方法的领域以外的经典情况。研究人员计划使用汉克尔算子的方法来解决所出现的问题，并期望发现新类型的解决方案，具有更复杂的波结构，与现有的孤子和辐射或周期（准周期）波列及其调制的简单性形成对比。这些结果将对应用产生影响，并与量子力学的基石薛定谔算子理论以及汉克尔和托普利茨算子理论有关，该项目由DMS应用数学计划和刺激竞争研究的既定计划（EPSCoR）共同资助。该奖项反映了NSF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。

The investigator will study nonlinear partial differential equations that are used in applications, to model phenomena in hydrodynamics, telecommunication, atmospheric sciences, nonlinear optics, plasma, and astrophysics. The specific focus is on the so-called completely integrable systems, an area of applied mathematics also known as soliton theory. Much of what is currently known concerns rapidly decaying or periodic initial conditions, but physics and applications call for more general non-standard situations: the thrust of this project is on understanding the effect of slower decay (or no decay) initial data. The project will improve the modeling and predicting capabilities for rogue waves, integrable turbulence, propagation of coherent structures in noisy media, tidal waves, and meteorological phenomena (i.e., morning glory). The project will also offer research and mentoring opportunities at both undergraduate and graduate levels. Recruitment efforts will be mindful to broaden research participation in mathematical science, as well as target students intending to become high school mathematics teachers. The project studies in the context of the Korteweg-de Vries equation the effect that initial data with slower decay (or no decay) at spatial plus infinity have on the solutions of completely integrable systems, that is partial differential equations that can be solved and analyzed by means of the inverse scattering transform (IST). Approaching the case of slower decay at plus infinity presents severe mathematical challenges in the applications of the IST, and its study is a major unsolved open question. Predicting long-time behavior of the solutions is difficult, as the well-known powerful Riemann-Hilbert machinery breaks down on such initial profiles. The main effort of this project will be devoted to extending the Riemann-Hilbert method outside of the realm of classical situations. The investigator plans to use the methods of Hankel operators to tackle the arising issues and expects to discover new types of solutions, with more complicated wave structure, in contrast to the simplicity of existing solitons and radiation, or periodic (quasi-periodic) wave trains and their modulations. The results will have implications for applications and be relevant to the theory of the Schrodinger operator, the cornerstone of quantum mechanics, and the theory of Hankel and Toeplitz operators, fundamental objects of operator theory.This project is jointly funded by the DMS Applied Mathematics Program and the Established Program to Stimulate Competitive Research (EPSCoR).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.