The modified KdV-Burgers equation
ut + (u3)x= μ uxx + β uxxx,
in which μ ≥ 0 and β are constant, is both dissipative anddispersive.
Moreover, the flux u3 is non-convex. Much can be learned from the structure
of solutions of initial value problems with Riemann initial data,
in which u(x,0) is piecewise constant with a single jump. When μ>0
the solutions are easily related to shock waves and rarefaction waves
for the conservation law ut+(u3)x=0. However, with μ=0,
the solutions involve dispersive shock waves (DSWs). I show how the two
cases are related, discuss the limit μ→ 0+, and demonstrate time
scales over which different wave structures appear. The construction
of the DSWs turns out to contain subtleties related to the presence of
undercompressive traveling waves for the μ>0 case, and to the construction
of shock-rarefaction wave solutions of the conservation law, due to the non-
This is joint work with Mark Hoefer (Univ. Colorado, Boulder) and Gennady El (Loughborough Univ.).