Dispersive approximations to nonlinear hyperbolic systems have
solitions that are highly oscillatory in regions where the solution of the
corresponding hyperbolic equations have a shock. As the dispersive
perturbation tends to zero, the solutons of the dispersive approximation
with fixed initial values tend weakly, but not strongly, to a limit.
These weak limits satisfy not the given hyperbolic equations but some
modifications of them. Some completely integrable cases can be analyzed
exactly. The modified eqations bear a resemblance to the appearance of
Reynolds stresses in turbulence theory.