# On universality of critical behaviour in the focusing nonlinear Schr\"odinger equation, elliptic umbilic catastrophe and the {\it tritronqu\'ee} solution to the Painlev\'e-I equation

## TL;DR

This paper demonstrates that the critical behavior near gradient catastrophe in the focusing nonlinear Schrödinger equation is approximately described by a special solution to the Painlevé-I equation, revealing universality in such nonlinear wave phenomena.

## Contribution

It establishes a connection between the critical behavior of the focusing nonlinear Schrödinger equation and the tritronquée solution of the Painlevé-I equation, highlighting a universal pattern.

## Key findings

- Critical behavior is described by Painlevé-I tritronquée solution.
- Universality of critical phenomena in nonlinear Schrödinger equations.
- Analytic initial data lead to Painlevé-I based asymptotics.

## Abstract

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.