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 (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 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.
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- March 2009
- DOI:
- 10.1007/s00220-008-0680-5
- arXiv:
- arXiv:0801.2326
- Bibcode:
- 2009CMaPh.286..979C
- Keywords:
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- Steep Descent Method;
- Hopf Equation;
- Hamiltonian Perturbation;
- Jump Matrix;
- Double Scaling Limit;
- Mathematical Physics;
- Mathematics - Analysis of PDEs
- E-Print:
- 30 pages