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

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

2009-11-30

arxiv.org/abs/0911.5686v1

Physics

Mathematical Physics

25 pages, 4 figures

Scientific paper

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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