  We consider several solitons moving in a slowly varying external field. We
show that the effective dynamics obtained by restricting the full Hamiltonian
to the finite dimensional manifold of $ N$-solitons (constructed when no
external field is present) provides a remarkably good approximation to the
actual soliton dynamics. That is quantified as an error of size $ h^2 $ where $
h $ is the parameter describing the slowly varying nature of the potential.
This also indicates that previous mathematical results of Holmer-Zworski for
one soliton are optimal. For potentials with unstable equilibria the Ehrenrest
time, $ \log(1/h)/h $, appears to be the natural limiting time for these
effective dynamics. We also show that the results of Holmer-Perelman-Zworski
for two mKdV solitons apply numerically to a larger number of interacting
solitons. We illustrate the results by applying the method with the external
potentials used in Bose-Einstein soliton train experiments of Strecker et. al.
