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.
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