  Using the complex Toda chain (CTC) as a model for the propagation of the
N-soliton pulse trains of the nonlinear Schrodinger (NLS) equation, we predict
the asymptotic behavior of these trains. The following asymptotic regimes are
stable: (i)~asymptotically free propagation of all N solitons; (ii)~bound state
regime where the N solitons may move quasi-equidistantly (QED); and
(iii)~various different combinations of (i) and (ii). For N=2 and 3 we
determine analytically the set of initial soliton parameters corresponding to
each of these regimes. We find excellent agreement between the solutions of CTC
and NLS for all regimes and propose realistic choices for the sets of
amplitudes, for which the solitons propagate QED to very large run lengths.
This is of importance for optical fiber communication.
