The KdV hierarchy: universality and a Painlevé transcendent

@article{Claeys2011TheKH,
  title={The KdV hierarchy: universality and a Painlev{\'e} transcendent},
  author={Tom Claeys and Tamara Grava},
  journal={International Mathematics Research Notices},
  year={2011},
  volume={2012},
  pages={5063-5099}
}
  • T. ClaeysT. Grava
  • Published 13 January 2011
  • Mathematics
  • International Mathematics Research Notices
We study the Cauchy problem for the Korteweg-de Vries (KdV) hierarchy in the small dispersion limit where $\e\to 0$. For negative analytic initial data with a single negative hump, we prove that for small times, the solution is approximated by the solution to the hyperbolic transport equation which corresponds to $\e=0$. Near the time of gradient catastrophe for the transport equation, we show that the solution to the KdV hierarchy is approximated by a particular Painlev\'e transcendent. This… 
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