Shock waves in semi-infinite photonic lattices

We investigate the existence, stability and propagation dynamics of shock waves existing in defocusing saturable media modulated by a semi-infinite photonic lattice. Such surface waves consist of modulationally stable pedestals in bulk media and decaying oscillatory tails in semi-infinite photonic lattices. Due to Bragg-type reflection, they are strongly localized at the edge of the optical lattice. The kink steepness, pedestal height and localization degree can be controlled by propagation constant, saturable degree and lattice depth. Two types of kinks, i.e., "out-of-phase" and two branches of "in-phase" shock waves are revealed. Out-of-phase shock waves are stable in a substantial part of their existence domain. While the lower-branch in-phase waves with low energy flow are stable in almost their whole existence domain, the higher-branch in-phase waves are completely unstable. We thus show the first example of stable in-phase shock waves in nonlinear optics. Our findings provide new insight into the dynamics of semi-localized nonlinear surface shock waves or kink solitons.