Universality for the Focusing Nonlinear Schr?dinger Equation at the Gradient Catastrophe Point: Rational Breathers and Poles of the Tritronquée Solution to Painlevé I

摘要

The semiclassical (zero-dispersion) limit of solutions $q=q(x,t,epsilon)$ to the one-dimensional focusing nonlinear Schr?dinger equation (NLS) is studied in a scaling neighborhood D of a point of gradient catastrophe ($x_0,t_0$). We consider a class of solutions, specified in the text, that decay as $|x| ightarrow infty$. The neighborhood D 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 q near the point of gradient catastrophe: (i) each spike has height $3|q{_0}(x_0,t_0)|$ and uniform shape of the rational breather solution to the NLS, scaled to the size ${cal O}(epsilon)$; (ii) the location of the spikes is determined by the poles of the tritronquée solution of the Painlevé I (P1) equation through an explicit map between D and a region of the Painlevé independent variable; (iii) if $(x,t)in D$ 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 tritronquée solution of P1. The relation with the conjecture of Dubrovin, Grava, and Klein about the behavior of solutions to the focusing NLS near a point of gradient catastrophe is discussed. 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.