Shock Waves in Dispersive Hydrodynamics with Nonconvex Dispersion

Abstract

Dissipationless hydrodynamics regularized by dispersion describe a number of physical media including water waves, nonlinear optics, and Bose-Einstein condensates. As in the classical theory of hyperbolic equations where a non-convex flux leads to non-classical solution structures, a non-convex linear dispersion relation provides an intriguing dispersive hydrodynamic analogue. Here, the fifth order Korteweg-de Vries (KdV) equation, also known as the Kawahara equation, a classical model for shallow water waves, is shown to be a universal model of Eulerian hydrodynamics with higher order dispersive effects. Utilizing asymptotic methods and numerical computations, this work classifies the long-time behavior of solutions for step-like initial data. For convex dispersion, the result is a dispersive shock wave (DSW), qualitatively and quantitatively bearing close resemblance to the KdV DSW. For non-convex dispersion, three distinct dynamic regimes are observed. For small jumps, a perturbed KdV DSW with positive polarity and orientation is generated, accompanied by small amplitude radiation from an embedded solitary wave leading edge, termed a radiating DSW or RDSW. For moderate jumps, a crossover regime is observed with waves propagating forward and backward from the sharp transition region. For jumps exceeding a critical threshold, a new type of DSW is observed we term a translating DSW or TDSW. The TDSW consists of a traveling wave that connects a partial, non-monotonic, negative solitary wave at the trailing edge to an interior nonlinear periodic wave. Its speed, a generalized Rankine-Hugoniot jump condition, is determined by the far-field structure of the traveling wave. The TDSW is resolved at the leading edge by a harmonic wavepacket moving with the linear group velocity. The non-classical TDSW exhibits features common to both dissipative and dispersive shock waves.