@article {2009,
title = {Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach},
journal = {Comm. Math. Phys. 286 (2009) 979-1009},
number = {arXiv.org;0801.2326},
year = {2009},
abstract = {We obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation in the small dispersion limit near the point of gradient catastrophe (x_c,t_c) for the solution of the dispersionless equation.\\nThe sub-leading term in this expansion is described by the smooth solution of a fourth order ODE, which is a higher order analogue to the Painleve I equation. This is in accordance with a conjecture of Dubrovin, suggesting that this is a universal phenomenon for any Hamiltonian perturbation of a hyperbolic equation. Using the Deift/Zhou steepest descent method applied on the Riemann-Hilbert problem for the KdV equation, we are able to prove the asymptotic expansion rigorously in a double scaling limit.},
doi = {10.1007/s00220-008-0680-5},
url = {http://hdl.handle.net/1963/2636},
author = {Tamara Grava and Tom Claeys}
}