The KdV hierarchy: universality and a Painlevé transcendent

Claeys, Tom;Grava, Tamara
(2011) International Mathematics Research Notices — Vol. 2011, n° 2011, p. 37 pages (2011)

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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 supports Dubrovins universality conjecture concerning the critical behavior of Hamiltonian perturbations of hyperbolic equations. We use the Riemann-Hilbert approach to prove our results.
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Claeys, T., & Grava, T. (2011). The KdV hierarchy: universality and a Painlevé transcendent. International Mathematics Research Notices, 2011(2011), 37 pages. https://doi.org/10.1093/imrn/rnr220 (Original work published 2011)