Nonlinear Wave Motion

非线性波动

• 批准号: 1712793
• 负责人: Mark Ablowitz
• 金额: $ 24.5万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-09-01 至 2021-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712793&HistoricalAwards=false
• 关键词: Nonlinear; Wave; Motion

Nonlinear wave motion is manifested in many natural phenomena and technological processes. Nonlinearity effects are most prominent for the high-power signals and large-magnitude waves: when one wave is superimposed on the other the waves are not just adding as would be true in the linear case but interact with each other. Such waves form complex wave patterns and have very important applications, including among others propagation of electromagnetic waves for fiber optics and lasers, shock waves in aerodynamics, and rogue waves in the ocean. Mathematical analysis of these phenomena is hampered by the nonlinearity of the governing equations. For nonlinear equations formulating a general theory is often unfeasible, and obtaining solutions requires a case-by-case study. This research effort focuses on solving a number of open problems that will substantially extend the ability of mathematicians to develop deeper understanding and explain the behavior of large-amplitude wave phenomena which arise widely in applications. The Inverse Scattering Transform (IST) method is one of the few analytical techniques that allows one to analyze and obtain solutions to a number of important equations in mathematical physics. IST will be used to obtain solutions and understand properties of new classes of nonlinear wave equations which exhibit PT (parity-time) symmetry properties.  IST can also be used to describe certain classes of localized solutions, termed lump solutions, which decay in all directions. The key properties of these solutions will be understood and they will be connected to the non-stationary Schrödinger equation. The research effort will also be directed to investigating a class of shock wave phenomena termed dispersive shock waves (DSWs). DSWs are shock waves which are regularized by dispersion, in contrast with standard shock waves which are regularized by dissipation. DSWs arise in many applications including water waves, Bose- Einstein condensates, nonlinear optics, etc. The theory of DSWs will be extended in order to obtain improved approximations to the underlying equations in one dimension, and a detailed multidimensional analysis will be developed.

非线性波动体现在许多自然现象和技术过程中。对于高功率信号和大幅度波来说，非线性效应最为突出：当一个波叠加在另一个波上时，这些波不仅像线性情况下那样相加，而且还会相互作用。这种波形成复杂的波型，具有非常重要的应用，其中包括光纤和激光的电磁波传播、空气动力学中的冲击波以及海洋中的异常波。这些现象的数学分析受到控制方程的非线性的阻碍。对于非线性方程，制定一般理论通常是不可行的，获得解需要具体情况具体分析。这项研究工作的重点是解决许多开放性问题，这些问题将大大扩展数学家的能力，以加深对应用中广泛出现的大幅波现象的理解和解释其行为。 逆散射变换 (IST) 方法是为数不多的分析技术之一，可以分析并获得数学物理中许多重要方程的解。 IST 将用于获得解并理解具有 PT（宇称时间）对称特性的新型非线性波动方程的特性。  IST 还可用于描述某些类别的局部解，称为块解，它们在所有方向上衰减。这些解的关键属性将被理解，并将它们与非平稳薛定谔方程联系起来。该研究工作还将致力于调查一类称为色散冲击波（DSW）的冲击波现象。 DSW 是通过色散规则化的冲击波，与通过耗散规则化的标准冲击波不同。 DSW 出现在许多应用中，包括水波、玻色-爱因斯坦凝聚、非线性光学等。DSW 的理论将得到扩展，以获得一维基础方程的改进近似，并且将开发详细的多维分析。

项目成果

Whitham equations and phase shifts for the Korteweg–de Vries equation

Whitham 方程和 Kortewegé Vries 方程的相移

• DOI: 10.1098/rspa.2020.0300
• 发表时间: 2020
• 期刊: Physical and Engineering Sciences
• 作者: Ablowitz, Mark J.;Cole, Justin T.;Rumanov, Igor
• 通讯作者: Rumanov, Igor

Integrable nonlocal asymptotic reductions of physically significant nonlinear equations

• DOI: 10.1088/1751-8121/ab0e95
• 发表时间: 2019-03
• 期刊: Journal of Physics A: Mathematical and Theoretical
• 作者: M. Ablowitz;Z. Musslimani

Solitons, the Korteweg-de Vries equation with step boundary values, and pseudo-embedded eigenvalues

孤子、具有步长边界值的 Korteweg-de Vries 方程和伪嵌入特征值

• DOI: 10.1063/1.5026332
• 发表时间: 2018
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Ablowitz, M. J.;Luo, X.-D.;Cole, J. T.
• 通讯作者: Cole, J. T.