Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws
We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion is known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is nonconvex. This review compares the structure of solutions of Riemann problems for a conservation law with nonconvex, cubic flux regularized by two different mechanisms: (1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and (2) a combination of diffusion and dispersion in the mKdV--Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs), and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both class...