Solitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit

Abstract

We study the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6u{u}_x+{\epsilon }^2{u}_{xxx}=0$ in a critical scaling regime where $x$ 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.

25 pages, 4 figures

References in corpus (3)

• Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach
• Universal distribution of random matrix eigenvalues near the "birth of a cut" transition
• Universality in the profile of the semiclassical limit solutions to the focusing Nonlinear Schroedinger equation at the first breaking curve