  Anomalous transport in one dimensional translation invariant Hamiltonian
systems with short range interactions, is shown to belong in general to the KPZ
universality class. Exact asymptotic forms for density-density and
current-current time correlation functions and their Fourier transforms are
given in terms of the Pr\"ahofer-Spohn scaling functions, obtained from their
exact solution for the Polynuclear growth model. The exponents of corrections
to scaling are found as well, but not so the coefficients. Mode coupling
theories developed previously are found to be adequate for weakly nonlinear
chains, but in need of corrections for strongly anharmonic interparticle
potentials.
