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

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.

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Correspondence to T. Claeys.

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Communicated by M. Aizenman

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Claeys, T., Grava, T. Universality of the Break-up Profile for the KdV Equation in the Small Dispersion Limit Using the Riemann-Hilbert Approach. Commun. Math. Phys. 286, 979 (2009). https://doi.org/10.1007/s00220-008-0680-5

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Keywords

  • Steep Descent Method
  • Hopf Equation
  • Hamiltonian Perturbation
  • Jump Matrix
  • Double Scaling Limit