  We numerically demonstrate the formation of the spatiotemporal version of the
so-called diffractive resonant radiation generated in waveguide arrays with
Kerr nonlinearity when a long pulse is launched into the system. The phase
matching condition for the diffractive resonant radiation that we have found
earlier for CW beams also works well in the spatiotemporal case. By introducing
a linear potential, one can introduce a continuous shift of the central
wavenumber of a linear pulse, whereas in the nonlinear case one can demonstrate
that the soliton self-wavenumber shift can be compensated by the emission of
diffractive resonant radiation, in a very similar fashion as it is done in
optical fibers. This work paves the way for designing unique optical devices
that generate spectrally broad supercontinua with a controllable directionality
by taking advantage of the combined physics of optical fibers and waveguide
arrays.
