A Haar-type Approximation and a New Numerical Schema for the Korteweg-de Vries Equation

TL;DR

This paper introduces a novel numerical method for solving the Korteweg-de Vries equation over large times, utilizing inverse scattering and wavelet-based approximations to improve resolution and efficiency.

Contribution

The authors develop a new recursive algorithm for the reflection coefficient using Haar wavelets, enhancing the numerical solution of the KdV equation.

Findings

High-resolution KdV solver achieved

Effective use of Haar wavelets for approximation

Potential for improved algorithmic efficiency

Abstract

We discuss a new numerical schema for solving the initial value problem for the Korteweg-de Vries equation for large times. Our approach is based upon the Inverse Scattering Transform that reduces the problem to calculating the reflection coefficient of the corresponding Schr\"odinger equation. Using a step-like approximation of the initial profile and a fragmentation principle for the scattering data, we obtain an explicit recursion formula for computing the reflection coefficient, yielding a high resolution KdV solver. We also discuss some generalizations of this algorithm and how it might be improved by using Haar and other wavelets.