Dispersive Hydrodynamics and Applications

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Tsunamis in the ocean and intense laser fiber optics are examples of waves whose speed of propagation depend on both the wave's wavelength and amplitude/intensity. The coincidence of these two properties in media that support waves is not limited to the ocean and fibers but is ubiquitous in nature, the lab, and technology. There is an emerging field of applied mathematics called dispersive hydrodynamics that explores the mathematical and physical ramifications of these two properties. This project will develop new mathematical solutions and models of nonlinear, dispersive waves in a variety of physical environments. The solutions will include solitons or solitary waves and dispersive shock waves, quintessential manifestations of waves whose speeds depend on wave amplitude and wavelength. An in-house laboratory shallow water experiment will also be constructed to realize and test the physical implications of the developed mathematics. Additional applications to fluid dynamics, nonlinear optics, and quantum fluids will be by-products of this research. Consequently, this project is truly interdisciplinary.Dispersive hydrodynamics has emerged as a unified mathematical framework for the description of multiscale nonlinear wave phenomena in dispersive media. Small-scale solitons/solitary waves and large-scale hydrodynamic waves (dispersive shock waves, rarefaction waves) are both solutions of nonlinear dispersive wave equations. This project weaves together research on integrable and nonintegrable dispersive partial differential equations, convex and nonconvex dispersive hydrodynamic systems, multidimensional waves, asymptotic analysis, and applications that include an in-house shallow water experiment. While the mathematics of solitons and that of dispersive shock waves have been extensively developed in isolation, this project explores the rich and physically motivated set of new mathematical problems related to soliton propagation in dispersive hydrodynamic flows. The scale separation inherent to solitons and dispersive hydrodynamic flows enables the asymptotic description of their interaction in a convenient and universal form in terms of a hyperbolic system of partial differential equations with a linearly degenerate field. Extending and applying this description to nonconvex, multidimensional, and attractive dispersive hydrodynamic flows are the primary aims of this project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.