Dispersive and Diffusive-Dispersive Shock Waves

Michael Shearer (North Carolina State University )

G.113, Alan Turing Building,

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-
convex flux.
This is joint work with Mark Hoefer (Univ. Colorado, Boulder) and Gennady El (Loughborough Univ.). 

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