An exact solution for the dispersive Alfven switch-on shock is found in the case of weak nonlinearity, weak dispersion, and weak damping. Two singular and apparently nonphysical solutions are found as well. The switch-on shock solution is shown to reduce, in the appropriate limits, to two documented special cases of the shock: one case being a description of the planar shock without dispersion and the other being an asymptotic description of the circularly polarized wave standing upstream from the shock in the dispersive case. The solution developed here provides the complete shock structure for either case. This exact solution may serve as a convenient basis for an analytical study of the stability of dispersive switch-on shocks.
Hamilton, Robert L. and Wu, C C., "An Exact Solution for the Dispersive Alfven Switch-on Shock" (2003). Faculty Publications - Department of Mathematics. 7.