@article {2012,
title = {The KdV hierarchy: universality and a Painleve transcendent},
journal = {International Mathematics Research Notices, vol. 22 (2012) , page 5063-5099},
number = {arXiv:1101.2602;},
year = {2012},
note = {This article was published in "International Mathematics Research Notices, vol. 22 (2012) , page 5063-5099},
publisher = {Oxford University Press},
abstract = {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.},
keywords = {Small-Dispersion limit},
url = {http://hdl.handle.net/1963/6921},
author = {Tom Claeys and Tamara Grava}
}
@article {2010,
title = {Painlev{\'e} II asymptotics near the leading edge of the oscillatory zone for the Korteweg-de Vries equation in the small-dispersion limit},
journal = {Comm. Pure Appl. Math. 63 (2010) 203-232},
number = {arXiv.org;0812.4142v1},
year = {2010},
publisher = {Wiley},
abstract = {In the small dispersion limit, solutions to the Korteweg-de Vries equation develop an interval of fast oscillations after a certain time. We obtain a universal asymptotic expansion for the Korteweg-de Vries solution near the leading edge of the oscillatory zone up to second order corrections. This expansion involves the Hastings-McLeod solution of the Painlev\\\\\\\'e II equation. We prove our results using the Riemann-Hilbert approach.},
doi = {10.1002/cpa.20277},
url = {http://hdl.handle.net/1963/3799},
author = {Tom Claeys and Tamara Grava}
}
@article {2010,
title = {Solitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit},
journal = {SIAM J. Math. Anal. 42 (2010) 2132-2154},
number = {SISSA;09/2010/FM},
year = {2010},
abstract = {We study the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\\\\epsilon^{2}u_{xxx}=0$ in a critical scaling regime where $x$ approaches the trailing edge of the region where the KdV solution shows oscillatory behavior. Using the Riemann-Hilbert approach, we obtain an asymptotic expansion for the KdV solution in a double scaling limit, which shows that the oscillations degenerate to sharp pulses near the trailing edge. Locally those pulses resemble soliton solutions of the KdV equation.},
doi = {10.1137/090779103},
url = {http://hdl.handle.net/1963/3839},
author = {Tamara Grava and Tom Claeys}
}
@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}
}