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

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 with the N-soliton solution of the Kadomtsev–Petviashvili equation. Experiments have made it clear that the system behaves like a coupled (an)harmonic oscillator on a discrete collapsed-state level. The streamlines obtained are derivatives of the error function as a function of the obtained Lax functional of the particle filaments dynamics induced by the (hypothetical) Calogero–Moser many-body system with elliptical potential and are the so-called Hermite functions. Hermite tried to introduce doubly periodic Hermite functions (the so-called Hermite problem) using coefficients related to the Weierstrass p-function. A solution-sensitive analysis of the incompressible viscid Navier–Stokes equation is performed using the Lamb vector. Cases with a meaningful potential-energy contribution require a particle interaction model with an N-soliton solution using a hierarchy-like solution of the Kadomtsev–Petviashvili equation. A three-soliton solution is emulated for the cylinder-wake problem. Our analytical results are put in perspective by comparison with two well-studied benchmark cases of fluid dynamics: the cylinder-wake problem and the driven-lid problem. The time-average velocity distribution (limit of streamline patterns) is consistent with published results and is enclosed in an asymmetrical lemniscate.