Shock Waves Without Dissipation

The Hopf equation u_t + uu_x = 0 is the simplest model of a breaking wave: smooth data steepen until the gradient blows up in finite time. Regularize with small viscosity and the vanishing-viscosity limit yields a classical shock. Regularize with small dispersion instead, as in the Korteweg–de Vries equation, and something very different happens. The singularity unfolds into an expanding, rank-ordered train of oscillations, with linear waves at one edge and solitons at the other. This is a dispersive shock wave (DSW). The oscillations persist as dispersion vanishes, so the limit exists only weakly. The right macroscopic description is Whitham modulation theory, a system of hyperbolic conservation laws for the wave’s slowly varying amplitude, wavenumber and mean. In a pleasing twist, a DSW is a rarefaction wave of the modulation equations.

I will build these ideas from the ground up and then follow them to three frontiers. First, I will regularize the Hopf equation by discretizing it with the textbook centered-difference scheme. Its Riemann problem is surprisingly rich, including an unsteady shock that radiates counterpropagating wavetrains at a common frequency and appears to be a new kind of modulation shock with no known continuum counterpart.  Second, in two dimensions, oblique DSWs can reflect regularly, as in the “matrix tide” recently observed on China’s Qiantang River. They can also undergo Mach reflection, where the stem is a genuine shock of the modulation equations and the wave at the wall grows up to eightfold. Finally, with random data, solitons form a gas governed by a kinetic equation. Splitting that gas into a deterministic condensate and a stochastic remainder gives an exactly solvable model of wave–mean field interaction, including stochastic DSWs.