Frontiers in Asymptotic Analysis for Integrable Nonlinear Waves

这个项目涉及的发展的半经典渐近技术的初值问题和初/边值问题的非线性波传播的情况下，感兴趣的方程是完全可积的，拥有一个Lax对表示和所有相应的数学框架。 具体问题包括：（i）中间长波方程的弱色散渐近性及其退化为Benjamin-Ono方程，（ii）半直线上半经典散焦非线性Schroedinger方程的混合初边值问题，（iii）半经典标度三波相互作用方程的初值问题，（iv）半经典修正的非线性Schroedinger方程的跨音速初值问题;（v）Painleve方程有理解的大次渐近性。 要解决这些问题，就需要为非局部和高维Riemann-Hilbert解析因式分解问题开发新的渐近分析工具。本项目涉及各种波的传播模型。 特别是，这些模型能够预测各种物理系统的行为，从携带高强度光脉冲的光纤到海洋，污染物可能被困在不同密度流体层之间的边界处。 我们的目标是提供实用的方法来准确地近似这些模型的解决方案，使他们的预测可以与实验进行比较，并最终用于代替昂贵的或侵入性的实验本身。 特别是，我们注意到，在2010年深水地平线石油泄漏等环境灾难之后，拥有准确和易于使用的方法来预测海洋液柱中污染物的位置变得越来越重要。

This project concerns the development of semiclassical asymptotic techniques for initial-value problems and initial/boundary-value problems of nonlinear wave propagation in the situation that the equation of interest is completely integrable, possessing a Lax pair representation and all of the corresponding mathematical framework. Specific problems to be addressed include (i) the weakly dispersive asymptotics of the intermediate long-wave equation and its degeneration to the Benjamin-Ono equation, (ii) mixed initial/boundary value problems for the semiclassical defocusing nonlinear Schroedinger equation on the half-line, (iii) initial-value problems for the semiclassically scaled 3-wave interaction equations, (iv) transsonic initial-value problems for the semiclassical modified nonlinear Schroedinger equation, and (v) large-degree asymptotics for rational solutions of Painleve equations. Solving these problems will require the development of new tools of asymptotic analysis for nonlocal and higher-dimensional Riemann-Hilbert analytic factorization problems.This project involves models for the propagation of waves of various sorts. In particular, the models are capable of making predictions about the behavior of physical systems ranging in diversity from optical fibers carrying high-intensity pulses of light to the ocean, where contaminants can become trapped at the boundary between layers of fluids of differing density. Our aim is to provide practical methods for accurately approximating the solutions of these models so that their predictions can be compared with experiments and ultimately used in place of expensive or invasive experiments themselves. In particular, we note that in the wake of environmental disasters like the Deepwater Horizon oil leak of 2010, it becomes increasingly important to have accurate and easy-to-use methods for predicting the location of contaminants in the fluid column of the ocean.