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

@article{Claeys2008UniversalityOT, title={Universality of the Break-up Profile for the KdV Equation in the Small Dispersion Limit Using the Riemann-Hilbert Approach}, author={Tom Claeys and Tamara Grava}, journal={Communications in Mathematical Physics}, year={2008}, volume={286}, pages={979-1009} }

AbstractWe 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 (xc, tc) for the solution of the dispersionless equation ut + 6uux = 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…

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