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Universality of the Break-up Profile for the KdV Equation in the Small Dispersion Limit Using the Riemann-Hilbert Approach

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Claeys, Tom [UCL] Grava, T

We obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation u(t) + 6uu(x) + epsilon(2)u(xxx) = 0, u(x, t = 0, epsilon) = u(0)(x), for epsilon small, near the point of gradient catastrophe (x(c), t(c)) for the solution of the dispersionless equation u(t) + 6uu(x) = 0. 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.

Document type Article de périodique (Journal article) – Article de recherche
Access type Accès restreint
Publication date 2009
Language Anglais
Journal information "Communications in Mathematical Physics" - Vol. 286, no. 3, p. 979-1009 (2009)
Peer reviewed yes
Publisher Springer ((Germany) Heidelberg)
issn 0010-3616
e-issn 1432-0916
Publication status Publié
Affiliation UCL - SC/MATH - Département de mathématique
Links
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Bibliographic reference Claeys, Tom ; Grava, T. Universality of the Break-up Profile for the KdV Equation in the Small Dispersion Limit Using the Riemann-Hilbert Approach. In: Communications in Mathematical Physics, Vol. 286, no. 3, p. 979-1009 (2009)
Permanent URL http://hdl.handle.net/2078.1/73372