Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws

Abstract

Abstract. 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 are 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 non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, 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 classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.

Modulation theory solution for nonlinearly resonant, fifth‐order Korteweg–de Vries, nonclassical, traveling dispersive shock waves

“…The solitary wave edge polarity, positive (wave of elevation) or negative (wave of depression), is determined by the combined effects of the nonlinear hydrodynamic flux curvature and linear dispersion curvature. 33 Until recently, most DSW research has focused on convex media, that is, nonlinear wave equations whose nonlinear hydrodynamic flux and linear dispersion relation have no inflection points. We refer to DSWs in convex media as classical in the sense that they generically resemble the canonical DSW first constructed by Gurevich and Pitaevskii for the Korteweg-de Vries (KdV) equation.…”

“…34 DSWs in the presence of nonconvex flux, however, can exhibit nonclassical structure, such as in the undercompressive DSW and the contact DSW. 33 For nonconvex linear dispersion, nonclassical DSW behavior can also emerge. In the absence of resonance, nonconvex linear dispersion can give rise to modulational instability in the vicinity of the DSW's linear wave edge, originally termed DSW implosion.…”

Stationary Expansion Shocks for a Regularized Boussinesq System

“…The nondimensional variables h, u represent the height of the water free surface above a flat horizontal bottom, and the depth-averaged horizontal component of the water velocity, respectively. System (1) is nonevolutionary, that is, not explicitly resolvable with respect to the time derivatives, a property that enables the possibility of new classes of solutions not generally observed in hyperbolic conservation laws and their evolutionary dispersive regularizations such as the Korteweg-de Vries equation, the defocusing nonlinear Schrödinger equation, and other equations exhibiting rich families of dispersive shock waves [3,4]. New solutions in the form of stationary, smooth, nonoscillatory expansion shocks were found in [5] for the Benjamin-Bona-Mahony (BBM) equation, that represents a unidirectional analog of the system (1).…”

“…The Boussinesq equations (1) are a convenient mathematical model in which to study expansion shocks for a system of dispersive equations. More broadly, the Boussinesq equations fall within the class of hyperbolic equations modified to incorporate dispersive terms, rather than dissipative terms, commonly referred to as dispersive hydrodynamic equations [4]. More familiar dispersive hydrodynamic solutions include oscillatory, compressive dispersive shock waves and expansive rarefaction waves.…”

Shock discontinuities: from classical to non-classical shocks

“…Jump discontinuities characterized by this unconventional property are termed non-classical under-compressive shocks and have been predicted also for creeping flows, Bertozzi, Münch and Shearer [2], suspensions of particles in liquids, Kluwick, Cox and Scheichl [24], constrained two-layer flows, Segin, Tilley and Kondic [61], dense gas acoustics, Scheichl and Kluwick [57,58] and the continuum theory of pedestrian flow, Colombo and Rosini [5]. Readers interested in a more rigerous mathematical theory of under-compressive non-classical shocks are referred to the review papers by LeFloch [40] and El, Hoefer, Shearer [11] which also include applications from other fields such as nonlinear elasticity and nonlinear optics. Due to the existence of a continuous inner structure such discontinuities areaccording to the third criterion listed in Sect.…”