An Airy beam as a self-similar solution to the problem of slit laser beam propagation in a linear medium and in a photorefractive crystal with diffusion nonlinearity

We have analysed self-similar solutions to the propagation problem of a slit beam with a plane wavefront in a linear medium and in a photorefractive crystal with diffusion nonlinearity. It is shown that in the latter case, despite the presence of the nonlinear term in the wave equation, the linear superposition principle holds true for the solutions of this class due to saturation. At the same time, the mirror symmetry violation of the wave equation for the transverse coordinate in the nonlinear case and the requirement to the spatial localisation modify one of the localised partial solutions (Airy beam) to the corresponding linear problem and prohibit the existence of other solutions of this class.