Nonlinear Wave Dynamics: Emergent Methods and Phenomena

这项研究项目将使用严格的渐近分析来回答以下三个方面的公开问题：(A)一个空间变量和一个时间变量的多分量可积系统初值问题的渐近行为，包括Burgers湍流的n波模型和马尔可夫n波模型。(B)了解和分类半经典Sine-Gordon方程和长时间非线性薛定谔方程等系统在边界区域的临界行为，并揭示相关Painleve函数的基本性质。(C)两个空间变量和一个时间变量模型的渐近分析，例如Davey-Stewartson II和Kadomtsev-Petviashvili II方程，需要采用d-bar方法进行逆散射。这三个项目将利用并进一步发展渐近分析、散射理论、正交多项式和随机矩阵研究的最新进展。非线性波动方程是流体动力学、固体物理和其他物理系统中能量传输的强大数学模型。所考虑的模型的具体应用包括宇宙学结构的演化、光缆和超导约瑟夫森结中波崩溃的开始、流体涡旋的积累和Hele-Shaw流。这些方程的解通常太复杂，不能完全笼统地描述，但在整个初始条件族的某些渐近极限下，显著特征往往变得明显。这种普适性表明该方程作为一个好的物理模型是合理的。在多组分系统中结合相互作用项的影响，分析关键转变以帮助完整地描述全局行为，并开发在两个空间维度研究系统所需的数学工具，将有助于更全面地了解一般非线性波动方程的行为及其作为物理模型的适用性。

This research project will use rigorous asymptotic analysis to answer open questions concerning the following three aspects of integrable nonlinear wave equations: (A) The asymptotic behavior of initial value problems for multicomponent integrable systems in one spatial and one temporal variable, including the n-wave model and the Markov n-wave model for Burgers turbulence. (B) Understanding and classifying critical behavior in boundary regions in systems such as the semiclassical sine-Gordon and long-time nonlinear Schrodinger equations, and uncovering fundamental properties of associated Painleve functions. (C) Asymptotic analysis of models in two spatial and one temporal variables, such as the Davey-Stewartson II and Kadomtsev-Petviashvili II equations, requiring the d-bar approach to inverse scattering. These three projects will make use of and further develop recent advances in asymptotic analysis, scattering theory, and the study of orthogonal polynomials and random matrices.Nonlinear wave equations are powerful mathematical models for energy transmission in fluid dynamics, solid state physics, and other physical systems. Specific applications of the models considered include the evolution of cosmological structures, the onset of wave collapse in fiber-optic cables and superconducting Josephson junctions, the accumulation of fluid vortices, and Hele-Shaw flow. Solutions to these equations are typically too complex to describe in full generality, yet salient features often become clear in certain asymptotic limits for entire families of initial conditions. This universality is an indication of the reasonableness of the equation as a good physical model. Incorporating the effect of interaction terms in multicomponent systems, analyzing critical transitions to help completely describe global behavior, and developing the mathematical tools necessary to study systems in two spatial dimensions will help build a fuller picture of the behavior of general nonlinear wave equations and their applicability as physical models.