Solitonic Asymptotics for the Korteweg–de Vries Equation in the Small Dispersion Limit

Tom Claeys, Tamara Grava

Research output: Contribution to journalArticle (Academic Journal)peer-review

23 Citations (Scopus)

Abstract

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.
Original language English 2132-2154 23 SIAM Journal on Mathematical Analysis 42 5 https://doi.org/10.1137/090779103 Published - Sep 2010