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

@article{Bertola2010UniversalityFT,
title={Universality for the Focusing Nonlinear Schr{\"o}dinger Equation at the Gradient Catastrophe Point: Rational Breathers and Poles of the Tritronqu{\'e}e Solution to Painlev{\'e} I},
author={Marco Bertola and Alexander Tovbis},
journal={Communications on Pure and Applied Mathematics},
year={2010},
volume={66}
}
• Published 11 April 2010
• Mathematics
• Communications on Pure and Applied Mathematics
The semiclassical (zero‐dispersion) limit of solutions $q=q(x,t,\epsilon)$ to the one‐dimensional focusing nonlinear Schrödinger 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 $|x| \rightarrow \infty$. 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…
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