Survey and Review

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The Survey and Review paper of this issue, “Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws,” by G. A. El, M. A. Hoefer, and M. Shearer, connects two well-known areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves, and uses a raft of applied mathematics tools: phase portrait analysis, multiple scales, averaging, etc. The theory of nonlinear dispersive waves has its roots in nineteenth century studies of water waves and gained an enormous prominence after the computer discovery of the soliton by Zabusky and Kruskal in 1965. This discovery led to the theory of integrable Hamiltonian partial differential equations with ramifications in many areas of mathematics, and also in mathematical physics, nonlinear optics, etc. The evolution of the theory of hyperbolic conservation laws has followed a similar pattern; while it started with Riemann's work on gas dynamics, it only came of age in the 1950s and 1960s with the contributions of Lax, Oleinik, and others. Regardless of the smoothness of the initial data, solutions of nonlinear conservation laws typically develop discontinuities and therefore have to be understood in a weak sense. To make matters more complicated, weak solutions are often not unique, and the solution with physical relevance has to be identified with the help of an entropy condition. A way to avoid these difficulties is to regularize the differential equations. Regularization is of course not only a matter of mathematical convenience: it brings to the equation “higher order” physical effects ignored in the conservation law formulation. The dissipative regularization of the inviscid Burgers equation \( u_t+(u^2/2)_x = 0\) results in the Burgers equation \( u_t+(u^2/2)_x = \epsilon u_xx\), \(\epsilon >0\). The inviscid shock that jumps from the upstream state \( u^-\) to the downstream state \(u^+\), \(u^->u^+\), is regularized to become a viscous, smooth traveling wave solution that shares the shock speed and upstream and downstream values. In situations without dissipation, regularization may arise from dispersion, as in the Korteweg--de Vries (KdV) equation \( u_t+(u^2/2)_x = \mu u_xxx\), and in that case the regularization of shock solutions leads to so-called dispersive shock waves, entities with a structure much richer than that of the dissipative traveling wave. The paper focuses on the modified KdV Burgers equation \( u_t+(u^3)_x = \epsilon u_xx+\mu u_xxx\), a universal asymptotic model where the corresponding conservation law \( u_t+(u^3)_x = 0\) has a nonconvex flow \( u^3\). As the paper shows, many remarkable phenomena appear due to the competition among nonconvexity, dissipation, and dispersion; these include undercompressive shock waves and double-wave complexes such as shock-rarefactions.