Universality for the focusing nonlinear Schroedinger equation at the gradient catastrophe point: Rational breathers and poles of the tritronquee solution to Painleve I

摘要

The semiclassical (zero-dispersion) limit of the one-dimensional focusingNonlinear Schroedinger equation (NLS) with decaying potentials is studied in afull scaling neighborhood D of the point of gradient catastrophe (x_0,t_0).This neighborhood contains the region of modulated plane wave (with rapid phaseoscillations), as well as the region of fast amplitude oscillations (spikes).In this paper we establish the following universal behaviors of the NLSsolutions near the point of gradient catastrophe: i) each spike has the height3|q_0(x_0,t_0,epsilon)| and uniform shape of the rational breather solution tothe NLS, scaled to the size O(epsilon); ii) the location of the spikes aredetermined by the poles of the tritronquee solution of the Painleve I (P1)equation through an explicit diffeomorphism between D and a region into thePainleve plane; iii) if (x,t) belongs to D but lies away from the spikes, theasymptotics of the NLS solution q(x,t,epsilon) is given by the plane waveapproximation q_0(x,t,epsilon), with the correction term being expressed interms of the tritronquee solution of P1. The latter result confirms theconjecture of Dubrovin, Grava and Klein about the form of the leading ordercorrection in terms of the tritronquee solution in the non-oscillatory regionaround (x_0,t_0). We conjecture that the P1 hierarchy occurs at higherdegenerate catastrophe points and that the amplitudes of the spikes are oddmultiples of the amplitude at the corresponding catastrophe point. Ourtechnique is based on the nonlinear steepest descent method for matrixRiemann-Hilbert Problems and discrete Schlesinger isomonodromictransformations.