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

Abstract

We argue that the critical behaviour near the point of ``gradient catastrophe" of the solution to the Cauchy problem for the focusing nonlinear Schrödinger 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 Painlevé-I equation.

32 pages, 10 figures