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

Dubrovin, B.; Grava, T.; Klein, C.

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

Authors

Dubrovin, B.

Grava, T.

Klein, C.

Description

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 $ iεψ_t +\frac{ε^2}2ψ_{xx}+ |ψ|^2 ψ=0$ with analytic initial data of the form $ψ(x,0;ε) =A(x) e^{\frac{i}ε S(x)}$ is approximately described by a particular solution to the Painlevé-I equation.
32 pages, 10 figures