Numerical integration of coupled Korteweg-de Vries System

TL;DR

The paper develops a general, stable finite‑difference framework for solving coupled KdV systems with arbitrary numbers of equations and coefficients, using a two‑step, three‑time‑level scheme with high‑order spatial discretization and an intermediate half‑step. It proves conditional stability and convergence, deriving grid‑size relations that guide timestep selection, and extends the analysis from a single KdV equation to the full vector cKdV system. The authors validate the method on the Hirota–Satsuma system by comparing to explicit multi‑soliton solutions and by checking a discrete $L^2$‑conservation law, showing accurate reproduction of soliton dynamics and robustness to parameter changes, near‑integrable perturbations, and non‑smooth initial data. The work provides a practical numerical tool for simulating Cauchy problems in multi‑component dispersive systems with potential applications in fluid dynamics and related fields.

Abstract

We introduce a numerical method for general coupled Korteweg-de Vries systems. The scheme is valid for solving Cauchy problems for arbitrary number of equations with arbitrary constant coefficients. The numerical scheme takes its legality by proving its stability and convergence which gives the conditions and the appropriate choice of the grid sizes. The method is applied to Hirota-Satsuma (HS) system and compared with its known explicit solution investigating the influence of initial conditions and grid sizes on accuracy. We also illustrate the method to show the effects of constants with a transition to non-integrable cases.