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

TL;DR

This paper numerically investigates the small dispersion limit of the Korteweg-de Vries equation, comparing solutions with asymptotic formulas across the entire space-time domain to understand their accuracy and matching.

Contribution

It provides a comprehensive numerical analysis of the KdV small dispersion limit and assesses the accuracy of various asymptotic solutions in different regions.

Findings

Asymptotic formulas match well with numerical solutions in specific regions.

Numerical results confirm the validity of asymptotic matching across the domain.

Quantitative comparison highlights the accuracy and limitations of asymptotic approximations.

Abstract

We study numerically the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\epsilon^{2}u_{xxx}=0$ for $\epsilon\ll1$ 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.