Universality for the focusing nonlinear Schrödinger equation at the gradient catastrophe point: rational breathers and poles of the Tritronquée solution to Painlevé I

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.