  Scalar conservation laws with non-convex fluxes have shock wave solutions
that violate the Lax entropy condition. In this paper, such solutions are
selected by showing that some of them have corresponding traveling waves for
the equation supplemented with dissipative and dispersive higher-order terms.
For a cubic flux, traveling waves can be calculated explicitly for linear
dissipative and dispersive terms. Information about their existence can be used
to solve the Riemann problem, in which we find solutions for some data that are
different from the classical Lax-Oleinik construction. We consider dispersive
terms of a BBM type and show that the calculation of traveling waves is
somewhat more intricate than for a KdV-type dispersion. The explicit
calculation is based upon the calculation of parabolic invariant manifolds for
the associated ODE describing traveling waves. The results extend to the
p-system of one-dimensional elasticity with a cubic stress-strain law.
