Transition from discrete to continuous Townes solitons in periodic media

We present a comprehensive analysis of how the properties of two-dimensional lattice (''discrete'') solitons in a Kerr medium are influenced by their peak intensity and width. We are able to quantitatively relate the Townes solution for solitons in a two-dimensional, homogeneous media to two distinct regimes of the lattice solitons: to narrow, high-intensity, highly nonlinear solitons and to broad, low-intensity, weakly nonlinear solitons, which experience the periodic potential as an effective homogeneous medium. Both regimes, although they support a different power flow and are affected by completely different diffraction dynamics, are thus traced back to the same physical phenomenon. They are separated by a range of unstable and stable solutions, directly caused by the periodicity of the lattice.