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

Claeys, Tom;Grava, T
(2009) Communications in Mathematical Physics — Vol. 286, n° 3, p. 979-1009 (2009)

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Abstract
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.
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Claeys, T., & Grava, T. (2009). Universality of the Break-up Profile for the KdV Equation in the Small Dispersion Limit Using the Riemann-Hilbert Approach. Communications in Mathematical Physics, 286(3), 979-1009. https://doi.org/10.1007/s00220-008-0680-5 (Original work published 2009)