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. The 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.
|Titolo:||Universality of the Break-up Proﬁle for the KdV Equation in the Small Dispersion Limit Using the Riemann-Hilbert Approach|
|Autori:||Claeys, T.; Grava, T.|
|Data di pubblicazione:||2009|
|Digital Object Identifier (DOI):||10.1007/s00220-008-0680-5|
|Appare nelle tipologie:||1.1 Journal article|