  Weak interactions of solitary waves in the generalized nonlinear
Schr\"{o}dinger equations are studied. It is first shown that these
interactions exhibit similar fractal dependence on initial conditions for
different nonlinearities. Then by using the Karpman-Solov'ev method, a
universal system of dynamical equations is derived for the velocities,
amplitudes, positions and phases of interacting solitary waves. These dynamical
equations contain a single parameter, which accounts for the different forms of
nonlinearity. When this parameter is zero, these dynamical equations are
integrable, and the exact analytical solutions are derived. When this parameter
is non-zero, the dynamical equations exhibit fractal structures which match
those in the original wave equations both qualitatively and quantitatively.
Thus the universal nature of fractal structures in the weak interaction of
solitary waves is analytically established. The origin of these fractal
structures is also explored. It is shown that these structures bifurcate from
the initial conditions where the solutions of the integrable dynamical
equations develop finite-time singularities. Based on this observation, an
analytical criterion for the existence and locations of fractal structures is
obtained. Lastly, these analytical results are applied to the generalized
nonlinear Schr\"{o}dinger equations with various nonlinearities such as the
saturable nonlinearity, and predictions on their weak interactions of solitary
waves are made.
