Numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions

Abstract

We study numerically the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6u{u}_x+{\epsilon }^2{u}_{xxx}=0$ for $\epsilon \ll 1$ and give a quantitative comparison of the numerical solution with various asymptotic formulae for small $\epsilon$ in the whole $(x,t)$-plane. The matching of the asymptotic solutions is studied numerically.

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• On critical behaviour in generalized Kadomtsev--Petviashvili equations
• Whitham modulation equations and application to small dispersion asymptotics and long time asymptotics of nonlinear dispersive equations
• Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart
• Numerical scattering for the defocusing Davey-Stewartson II equation for initial data with compact support
• Numerical Approach to Painlevé Transcendents on Unbounded Domains
• On the tritronquée solutions of P${}_I^2$
• Multi-domain spectral approach for the Hilbert transform on the real line