  We consider dispersive shock wave to the focusing nonlinear Schr\"odinger
equation generated by a discontinuous initial condition which is periodic or
quasi-periodic on the left semi-axis and zero on the right semi-axis. As an
initial function we use a finite-gap potential of the Dirac operator given in
an explicit form through hyper-elliptic theta-functions. The paper aim is to
study the long-time asymptotics of the solution of this problem in a vicinity
of the leading edge, where a train of asymptotic solitons are generated. Such a
problem was studied in \cite{KK86} and \cite{K91} using Marchenko's inverse
scattering technics. We investigate this problem exceptionally using the
Riemann-Hilbert problems technics that allow us to obtain explicit formulas for
the asymptotic solitons themselves that in contrast with the cited papers where
asymptotic formulas are obtained only for the square of absolute value of
solution. Using transformations of the main RH problems we arrive to a model
problem corresponding to the parametrix at the end points of continuous
spectrum of the Zakharov-Shabat spectral problem. The parametrix problem is
effectively solved in terms of the generalized Laguerre polynomials which are
naturally appeared after appropriate scaling of the Riemann-Hilbert problem in
a small neighborhoods of the end points of continuous spectrum. Further
asymptotic analysis give an explicit formula for solitons at the edge of
dispersive wave. Thus, we give the complete description of the train of
asymptotic solitons: not only bearing envelope of each asymptotic soliton, but
its oscillating structure are found explicitly. Besides the second term of
asymptotics describing an interaction between these solitons and oscillating
background is also found. This gives the fine structure of the edge of
dispersive shock wave.
