A Mathematical Justification of the Momentum Density Function Associated to the KdV Equation

TL;DR

This paper provides a mathematical justification for an approximate momentum density linked to the KdV equation, comparing it to the physical momentum density from Euler equations, with error estimates based on the long-wave parameter.

Contribution

It introduces and justifies an approximate momentum density for the KdV equation and quantifies its deviation from the true physical momentum density.

Findings

Approximate momentum density is mathematically derived for the KdV equation.

The difference between approximate and physical momentum densities is estimated in terms of the long-wave parameter.

The results validate the use of the KdV model for long-wave approximations in fluid dynamics.

Abstract

Consideration is given to the KdV equation as an approximate model for long waves of small amplitude at the free surface of an inviscid fluid. It is shown that there is an approximate momentum density associated to the KdV equation, and the difference between this density and the physical momentum density derived in the context of the full Euler equations can be estimated in terms of the long-wave parameter.