CAREER: Solitary Waves and Wavetrains in Dispersive Media

这个跨学科的项目研究孤立波和某些非线性，色散偏微分方程的解决方案，利用分析，数值和实验方法相结合。 其目的是提供一个准确的，实用的，详细的描述这些相干波结构，是有用的应用在纳米磁学和无耗散，色散介质的流体动力学。 最近的实验，时间周期，本地化的解决方案的修改后的Landau-Lifshitz方程，其中包括一个自旋转移力矩强迫项，以补偿阻尼将计算和研究分析使用渐近方法。 这些磁液滴孤子及其推广具有潜在的技术应用。另一个研究前沿将涉及调制非线性波列或色散冲击波（DSW）。 DSW的产生代表了一种通用的机制，以解决色散介质，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，粘性变形流体中的流体动力学奇点。 理论研究将集中在现代DSW理论的核心问题，包括真正的非线性/严格双曲性的调制方程，多维DSW和稳定性的损失。 主要研究者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，并与理论进行比较。当波的特性本质上取决于波的振幅时，可能会出现新的物理行为。 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冲击波。 自旋电子学包括利用电子的自旋和电荷来开发信息传输、处理和存储的努力，例如，延续摩尔定律（每单位面积的晶体管数量每十八个月增加一倍）超越目前的技术限制。 在这个项目中要研究的纳米磁体中的空间局域自旋激发具有为未来自旋电子学应用带来巨大希望的功能。 通常认为，当抛射体在空气中超过声速时形成粘性冲击波。在这个项目中研究的分散冲击波是一个非常不同的类型，缺乏耗散和实现的扩展，振荡波列。 将开发一个流体实验来仔细测量这些冲击特性，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。 在北卡罗来纳州自然科学博物馆的演讲将为公众提供接触这一迷人领域的机会。

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.