The KdV hierarchy: universality and a Painleve transcendent

Details The KdV hierarchy: universality and a Painleve transcendent The KdV hierarchy: universality and a Painleve transcendent

2011-01-13

arxiv.org/abs/1101.2602v1

Physics

Mathematical Physics

Scientific paper

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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