Collapse of the two-dimensional spatial soliton in a parabolic-index material

Using a scale transformation, I relate the solution of a beam propagating in a parabolic-index Kerr material to its solution in a homogeneous Kerr material. This technique is then applied to prove that a two-dimensional cw beam that propagates as a spatial soliton in a homogeneous Kerr medium would collapse if instead it were propagating in a parabolic-index material and that its mode of collapse is different from that of a low-intensity beam propagating in the same parabolic medium.