  We use multiscale perturbation theory in conjunction with the inverse
scattering transform to study the interaction of a number of solitons of the
cubic nonlinear Schroedinger equation under the influence of a small correction
to the nonlinear potential. We assume that the solitons are all moving with the
same velocity at the initial instant; this maximizes the effect each soliton
has on the others as a consequence of the perturbation. Over the long time
scales that we consider, the amplitudes of the solitons remain fixed, while
their center of mass coordinates obey Newton's equations with a force law for
which we present an integral formula. For the interaction of two solitons with
a quintic perturbation term we present more details since symmetries -- one
related to the form of the perturbation and one related to the small number of
particles involved -- allow the problem to be reduced to a one-dimensional one
with a single parameter, an effective mass. The main results include
calculations of the binding energy and oscillation frequency of nearby solitons
in the stable case when the perturbation is an attractive correction to the
potential and of the asymptotic "ejection" velocity in the unstable case.
Numerical experiments illustrate the accuracy of the perturbative calculations
and indicate their range of validity.
