Title: 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 # Issue Date: 2009 Publisher: Springer-Verlag Heidelberg Series Title: Communications in Mathematical Physics vol:286 issue:3 pages:979-1009 Abstract: We obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation $$u_t+6uu_x+\epsilon^{2}u_{xxx}=0,\quad u(x,t=0,\epsilon)=u_0(x),$$ 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 Painlevé 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. ISSN: 0010-3616 Publication status: published KU Leuven publication type: IT Appears in Collections: Analysis Section