A note on N-soliton solutions for the viscid incompressible Navier-Stokes differential equation

TL;DR

This paper explores how higher-order derivatives in the viscid incompressible Navier-Stokes equation lead to N-soliton solutions, connecting fluid dynamics with integrable systems like the Korteweg-de Vries-Burgers equation.

Contribution

It introduces the concept of N-soliton solutions for the Navier-Stokes equation via higher-order derivatives and relates them to known soliton solutions of integrable equations.

Findings

Derivation of N-soliton solutions for Navier-Stokes.

Connection between Navier-Stokes and Korteweg-de Vries-Burgers equation.

Use of Weierstrass p-function in soliton solutions.

Abstract

Repetitive curling of the incompressible viscid Navier-Stokes differential equation leads to a higher-order diffusion equation. Substituting this equation into the Navier-Stokes differential equation transposes the latter into the Korteweg-de Vries-Burgers equation with the Weierstrass p-function as the soliton solution. However, a higher-order derivative of the studied variable produces the so-called N-soliton solution, which is comparable to the N-soliton solution of the Kadomtsev-Petviashvili equation.