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}
}
  • T. ClaeysT. Grava
  • Published 15 January 2008
  • Mathematics
  • Communications in Mathematical Physics
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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