On Universality of Critical Behavior in the Focusing Nonlinear Schrödinger Equation, Elliptic Umbilic Catastrophe and the Tritronquée Solution to the Painlevé-I Equation

We argue that the critical behavior near the point of "gradient catastrophe" of the solution to the Cauchy problem for the focusing nonlinear Schrodinger equation \(i\epsilon \varPsi _{t}+\frac{\epsilon^{2}}{2}\varPsi _{xx}+|\varPsi |^{2}\varPsi =0\) , e ≪1, with analytic initial data of the form \(\varPsi (x,0;\epsilon)=A(x)e^{\frac{i}{\epsilon}S(x)}\) is approximately described by a particular solution to the Painleve-I equation