We study the propagation of high frequency oscillations for one dimensional semi-linear hyperbolic systems with small parabolic perturbations. We obtain a new degenerate parabolic system for the profile, and valid an asymptotic development in the spirit of Joly, Métivier and Rauch.
Stéphane Junca . "Geometric optics with critical vanishing viscosity for one-dimensional semilinear initial value problems." Rev. Mat. Iberoamericana 24 (2) 549 - 566, July, 2008.