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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

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:
05.06.07
Published:
05.06.07
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