5 research outputs found
Solitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit
We study the small dispersion limit for the Korteweg-de Vries (KdV) equation
in a critical scaling regime where
approaches the trailing edge of the region where the KdV solution shows
oscillatory behavior. Using the Riemann-Hilbert approach, we obtain an
asymptotic expansion for the KdV solution in a double scaling limit, which
shows that the oscillations degenerate to sharp pulses near the trailing edge.
Locally those pulses resemble soliton solutions of the KdV equation.Comment: 25 pages, 4 figure
Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach
We obtain an asymptotic expansion for the solution of the Cauchy problem for
the Korteweg-de Vries (KdV) equation in the small dispersion limit near the
point of gradient catastrophe (x_c,t_c) for the solution of the dispersionless
equation.
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 Painleve 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.Comment: 30 page