  The semiclassical (zero-dispersion) limit of the one-dimensional focusing
Nonlinear Schroedinger equation (NLS) with decaying potentials is studied in a
full scaling neighborhood D of the point of gradient catastrophe (x_0,t_0).
This neighborhood contains the region of modulated plane wave (with rapid phase
oscillations), as well as the region of fast amplitude oscillations (spikes).
In this paper we establish the following universal behaviors of the NLS
solutions near the point of gradient catastrophe: i) each spike has the height
3|q_0(x_0,t_0,epsilon)| and uniform shape of the rational breather solution to
the NLS, scaled to the size O(epsilon); ii) the location of the spikes are
determined by the poles of the tritronquee solution of the Painleve I (P1)
equation through an explicit diffeomorphism between D and a region into the
Painleve plane; iii) if (x,t) belongs to D but lies away from the spikes, the
asymptotics of the NLS solution q(x,t,epsilon) is given by the plane wave
approximation q_0(x,t,epsilon), with the correction term being expressed in
terms of the tritronquee solution of P1. The latter result confirms the
conjecture of Dubrovin, Grava and Klein about the form of the leading order
correction in terms of the tritronquee solution in the non-oscillatory region
around (x_0,t_0). We conjecture that the P1 hierarchy occurs at higher
degenerate catastrophe points and that the amplitudes of the spikes are odd
multiples of the amplitude at the corresponding catastrophe point. Our
technique is based on the nonlinear steepest descent method for matrix
Riemann-Hilbert Problems and discrete Schlesinger isomonodromic
transformations.
