Riemann problems and dispersive shocks in self-focusing media

The dynamical behavior resulting from an initial discontinuity in focusing media is investigated using a combination of numerical simulations and Whitham modulation theory for the focusing nonlinear Schrödinger equation. Initial conditions with a jump in either or both the amplitude and the local wave number are considered. It is shown analytically and numerically that the space-time plane divides into expanding domains in which the solution is described by a slow modulation of genus-0, genus-1 or genus-2 solutions, their precise arrangement depending on the specifics of the initial datum.