  Previous studies have shown that fractal scatterings in weak interactions of
solitary waves in the generalized nonlinear Schr\"odinger equations are
described by a universal second-order separatrix map. In this paper, this
separatrix map is analyzed in detail, and hence a complete characterization of
fractal scatterings in these weak interactions is obtained. In particular,
scaling laws of these fractals are derived analytically for different initial
conditions, and these laws are confirmed by direct numerical simulations. In
addition, an analytical criterion for the occurrence of fractal scatterings is
given explicitly.
