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

Boris Dubrovin, Tamara Grava, and Christian Klein

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Submission date: 05. Jun. 2007
Pages: 41
published in: Journal of nonlinear science, 19 (2009) 1, p. 57-94 
DOI number (of the published article): 10.1007/s00332-008-9025-y
Bibtex
Keywords and phrases: nonlinear Schrödinger equation, gradient catastrophe, Painlevé equation
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Abstract:
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 , , with analytic initial data of the form is approximately described by a particular solution to the Painlevé-I equation.