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Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach

Authors
  • Claeys, Tom
  • Grava, Tamara
Type
Published Article
Publication Date
Jan 15, 2008
Submission Date
Jan 15, 2008
Identifiers
DOI: 10.1007/s00220-008-0680-5
Source
arXiv
License
Unknown
External links

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. 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.

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