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Journal of Nonlinear Science

, Volume 19, Issue 1, pp 57–94 | Cite as

On Universality of Critical Behavior in the Focusing Nonlinear Schrödinger Equation, Elliptic Umbilic Catastrophe and the Tritronquée Solution to the Painlevé-I Equation

  • B. DubrovinEmail author
  • T. Grava
  • C. Klein
Open Access
Article

Abstract

We argue that the critical behavior near the point of “gradient catastrophe” of the solution to the Cauchy problem for the focusing nonlinear Schrödinger equation \(i\epsilon \varPsi _{t}+\frac{\epsilon^{2}}{2}\varPsi _{xx}+|\varPsi |^{2}\varPsi =0\) , ε 1, with analytic initial data of the form \(\varPsi (x,0;\epsilon)=A(x)e^{\frac{i}{\epsilon}S(x)}\) is approximately described by a particular solution to the Painlevé-I equation.

Keywords

Nonlinear Schrödinger equation Gradient catastrophe Painlevé equations 

Mathematics Subject Classification (2000)

35Q55 33E17 37K05 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.SISSATriesteItaly
  2. 2.Max-Planck InstituteLeipzigGermany

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