  We present a detailed analysis of the solution of the focusing nonlinear
Schr\"odinger equation with initial condition $\psi(x,0)=N {\rm sech}(x)$ in
the limit $N\to\infty$. We begin by presenting new and more accurate numerical
reconstructions of the $N$-soliton by inverse scattering (numerical linear
algebra) for N=5, 10, 20, and 40. We then recast the inverse-scattering problem
as a Riemann-Hilbert problem and provide a rigorous asymptotic analysis of this
problem in the large-N limit. For those $(x,t)$ where results have been
obtained by other authors, we improve the error estimates from $O(N^{-1/3})$ to
$O(N^{-1})$. We also analyze the Fourier power spectrum in this regime and
relate the results to the optical phenomenon of supercontinuum generation. We
then study the $N$-soliton for values of $(x,t)$ where analysis has not been
carried out before, and we uncover new phenomena. The main discovery of this
paper is the mathematical mechanism for a secondary caustic (phase transition),
which turns out to differ from the mechanism that generates the primary
caustic. The mechanism for the generation of the secondary caustic depends
essentially on the discrete nature of the spectrum for the $N$-soliton, and
more significantly, cannot be recovered from an analysis of an ostensibly
similar Riemann-Hilbert problem in the conditions of which a certain formal
continuum limit is taken on an {\em ad hoc} basis
