CAREER: Solitary Waves and Wavetrains in Dispersive Media

这个跨学科的项目利用分析、数值和实验方法的结合来研究某些非线性、色散偏微分方程的孤立波和波列解。目的是提供这些相干波结构的准确、实用、详细的描述，这对纳米磁学和无耗散、色散介质的流体动力学的应用是有用的。基于最近的实验，我们将使用渐近方法计算并研究一个修正的Landau-Lifshitz方程的时间周期局部解，该方程包含一个自旋传递扭矩强迫项来补偿阻尼。这些磁液滴孤子及其推广具有技术应用潜力。另一个研究前沿将涉及调制非线性波串或色散激波（DSWs）。dsw的产生代表了一种解决色散介质（如水波、等离子体、光学、界面流体、玻色-爱因斯坦凝聚体和两相粘性变形流体）中水动力奇点的通用机制。理论研究将集中于现代DSW理论的核心问题，包括调制方程中真正非线性/严格双曲性的损失、多维DSW和稳定性。首席研究员将与本科生和研究生合作，进行涉及粘性流体管道系统的实验，以便对DSW特性进行定量测量，以便与理论进行比较。当波的性质本质上取决于波的振幅时，就会产生新的物理行为。从应用的角度来看，本项目包括两类非线性波：纳米磁性中的孤立波和色散介质中的激波。自旋电子学包括利用电子的自旋和电荷来发展信息传输、处理和存储的努力，例如，超越目前的技术限制，继续遵循摩尔定律（每18个月单位面积上晶体管的数量翻一番）。本课题研究的纳米磁体的空间局域自旋激发具有未来自旋电子学应用前景广阔的特点。当抛射物在空气中超过声速时形成的粘性激波是人们普遍理解的。本项目研究的色散激波是一种非常不同的类型，缺乏耗散，以膨胀的振荡波串的形式实现。将开发一个流体实验来仔细测量这些激波特性，同时也被用作教育，推广和演示工具，为年轻的数学家提供非线性波的直接经验。在北卡罗来纳自然科学博物馆的演讲将为公众提供接触这个迷人领域的机会。

This interdisciplinary project examines solitary wave and wavetrain solutions of certain nonlinear, dispersive partial differential equations utilizing a combination of analytical, numerical, and experimental approaches. The aim is to provide an accurate, practical, detailed description of these coherent wave structures that is useful for applications in nanomagnetism and the fluid dynamics of dissipationless, dispersive media. Motivated by recent experiments, time-periodic, localized solutions of a modified Landau-Lifshitz equation that incorporates a spin transfer torque forcing term to compensate damping will be computed and studied analytically using asymptotic methods. These magnetic droplet solitons and their generalizations have potential for technological applications. Another research front will be developed involving modulated nonlinear wavetrains or dispersive shock waves (DSWs). The generation of DSWs represents a universal mechanism to resolve hydrodynamic singularities in dispersive media such as water waves, plasma, optics, interfacial fluids, Bose-Einstein condensates, and two-phase, viscously deformable fluids. Theoretical studies will focus upon central questions of modern DSW theory including loss of genuine nonlinearity/strict hyperbolicity in the modulation equations, multi-dimensional DSWs, and stability. In collaboration with undergraduate and graduate students, the principal investigator will undertake experiments involving a viscous fluid conduit system in order to make quantitative measurements of DSW properties for comparison with theory.When wave properties intrinsically depend upon the wave amplitude, novel physical behavior can arise. With a view toward applications, this project encompasses two classes of these nonlinear waves: solitary waves in nanomagnetism and shock waves in dispersive media. Spintronics encompasses the effort to develop information transport, processing, and storage using the electron's spin in addition to its charge, for example, to continue Moore's Law (the doubling of the number of transistors per unit area every eighteen months) beyond present technological limitations. The spatially localized spin excitations in a nanomagnet to be studied in this project possess features that hold great promise for future spintronic applications. Viscous shock waves that form when a projectile exceeds the speed of sound in air are commonly understood. The dispersive shock waves studied in this project are of a very different type, lacking dissipation and realized as expanding, oscillatory wavetrains. A fluid experiment will be developed to carefully measure these shock properties while also being used as an educational, outreach, and demonstration tool, providing young mathematicians direct experience with nonlinear waves. Presentations at the North Carolina Museum of Natural Sciences will provide the general public with exposure to this fascinating field.