Stationary shock-like transition fronts in dispersive systems

We show that, contrary to popular belief, lower order dispersive regularization of hyperbolic systems does not exclude the development of the localized shock-like transition fronts. To guide the numerical search of such solutions, we generalize Rankine-Hugoniot relations to cover the case of higher order dispersive discontinuities and study their properties in an idealized case of a transition between two periodic wave trains with different wave lengths. We present evidence that smoothed stationary fronts of this type are numerically stable in the case when regularization is temporal and one of the adjacent states is homogeneous. In the zero dispersion limit such shock-like transition fronts, that are not traveling waves and apparently require for their description more complex anzats, evolve into traveling wave type jump discontinuities