On universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the {\it tritronquée} solution to the Painlevé-I equation

B. Dubrovin, T. Grava, C. Klein

Published 2007-04-04, updated 2007-05-15Version 3

We argue that the critical behaviour near the point of ``gradient catastrophe" of the solution to the Cauchy problem for the focusing nonlinear Schr\"odinger equation $ i\epsilon \psi_t +\frac{\epsilon^2}2\psi_{xx}+ |\psi|^2 \psi =0$ with analytic initial data of the form $\psi(x,0;\epsilon) =A(x) e^{\frac{i}{\epsilon} S(x)}$ is approximately described by a particular solution to the Painlev\'e-I equation.