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


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.


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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