Solitonic Asymptotics for the Korteweg-De Vries Equation in the small dispersion limit

Claeys, Tom;Grava, T
(2010) SIAM Journal on Mathematical Analysis — Vol. 42, n° 5, p. 2132-2154 (2010)

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We study the small dispersion limit for the Korteweg-de Vries (KdV) equation u(t) + 6uu(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.
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Claeys, T., & Grava, T. (2010). Solitonic Asymptotics for the Korteweg-De Vries Equation in the small dispersion limit. SIAM Journal on Mathematical Analysis, 42(5), 2132-2154. https://doi.org/10.1137/090779103 (Original work published 2010)