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MiS Preprint
54/2007

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

Abstract

We argue that the critical behaviour near the point of "gradient catastroph" 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$, $\epsilon\ll 1$, 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 Painlevé-I equation.

Received:
Jun 5, 2007
Published:
Jun 5, 2007
Keywords:
nonlinear Schrödinger equation, gradient catastrophe, Painlevé equation

Related publications

inJournal
2009 Journal Open Access
Boris Dubrovin, Tamara Grava and Christian Klein

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

In: Journal of nonlinear science, 19 (2009) 1, pp. 57-94