ABSTRACT

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.

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