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.