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

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

2007-04-04

arxiv.org/abs/0704.0501v3

Mathematics

Analysis of PDEs

32 pages, 10 figures

Scientific paper

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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