Dispersive Hydrodynamics: the Mathematics of Dispersive Shock Waves and Applications

Dispersive hydrodynamics is the domain of applied mathematics and physics concerned with fluid motion in which internal friction, e.g., viscosity, is negligible relative to wave dispersion. In conservative media such as superfluids, optical materials, and water waves, nonlinearity has the tendency to engender wavebreaking that is mitigated by dispersion. The mathematical framework for such media can often be described by hyperbolic systems of partial differential equations with conservative, dispersive corrections that play a fundamental role in the dynamics. Generically, the result of nonlinearity and dispersion is a multiscale, unsteady, coherent wave structure called a dispersive shock wave or DSW. Over long time scales, multiple wavebreaking events or, in focusing media, modulational instability can lead to the development of soliton (strong) turbulence. This meeting brought together an international collection of mathematicians and physicists in order to identify common interests and emerging problems involving DSWs, soliton turbulence, and their mathematical description. This field of research has origins in soliton theory, conservation laws, and fluid dynamics. In 1965, the seminal computational work of Zabusky and Kruskal [1] demonstrated the existence of soliton solutions to the Korteweg-de Vries (KdV) equation through a process of nonlinear wavebreaking. That same year, Whitham introduced a general asymptotic approach to study modulated periodic nonlinear dispersive waves [2, 3]. Both of these works considered conservative, nonlinear, dispersive wave problems. Again in 1965, although within the context of a different field of research, Glimm’s fundamental work on hyperbolic conservation laws [4] contributed to the rapid growth in understanding of this field, see, e.g. [5]. The marriage of dispersive nonlinear waves and hyperbolic conservation laws in the context of dispersive hydrodynamics was initiated in 1974 at the hands of Gurevich and Pitaevskii [6] through their study of a Riemann problem regularized by dispersion in the KdV equation. The resulting dispersive shock waves were understood utilizing Whitham’s modulation equations, later shown to describe the weak, zero dispersion limit of the KdV equation by Lax, Levermore, and Venakides [8, 9]. The KdV Whitham modulation equations are now known to be strictly hyperbolic and genuinely nonlinear [10], highlighting the deep connections between conservation laws and dispersive nonlinear waves in dispersive hydrodynamics. Another fundamental work was on averaging of multiphase solutions to the KdV equation utilizing finite gap theory [7]. An essential aspect of dispersive hydrodynamics is its physical realization. Laboratory measurements of DSWs were undertaken in the context of undular bores in shallow water waves by Favre in 1935 [11].