A note on N-soliton solutions for the viscid incompressible Navier-Stokes differential equation (2506.23697v1)

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.

• The paper derives N-soliton solutions for the 3D Navier-Stokes equations by recursively applying the curl operator.
• It transforms the Navier-Stokes equation into a KdV-Burgers framework using connections to the Weierstrass p-function and Madelung's fluid model.
• The study demonstrates a non-perturbative method for generating smooth, global soliton solutions with implications for classic fluid dynamics benchmarks.

N-Soliton Solutions for the Viscid Incompressible Navier-Stokes Equation

This paper presents an analytical approach to solving the 3D incompressible Navier-Stokes equations by constructing N-soliton solutions. The method involves recursively applying the curl operator to the Navier-Stokes equations, leading to a hierarchy of higher-order diffusion equations. These equations are then related to the Korteweg-De Vries-Burgers equation, which admits solutions involving the Weierstrass p-function. The author claims that higher-order derivatives of the studied variable yield N-soliton solutions comparable to those of the Kadomtsev-Petviashvili equation. This approach is positioned as a non-perturbative method for obtaining smooth, global solutions without resorting to simplifications or numerical approximations.

Theoretical Framework

The paper begins by outlining the roadmap for solving the incompressible viscid Navier-Stokes equations. By taking two curls of the vortex transport equation, a diffusion equation for higher derivatives of vorticity vectors is derived. This manipulation leads to the transposition of the Navier-Stokes equation into a Korteweg-De Vries-Burgers equation. The author leverages Madelung's fluid concept to couple a nonlinear Schrödinger-like equation to the KdV equation, thereby linking the initial-value problem of the Navier-Stokes equation to soliton solutions. The triviality of the Navier-Stokes solution is attributed to its nature as an initial-boundary problem, which is interpreted as a moving boundary or singularity of the Lax functional.

Mathematical Formalism

The core of the paper involves the transition of the Navier-Stokes equation to the Korteweg-De Vries-Burgers equation using the Lamb vector. The Lamb vector facilitates the decomposition of fluid dynamical flows into divergence-free and curl-free components. Repeatedly applying the curl operator to the Navier-Stokes equation results in a diffusion equation with known solutions. This process leads to the Korteweg-De Vries-Burgers equation as a byproduct. The Weierstrass p-function is introduced as a solution for flow problems with a meaningful potential-energy contribution, relating these problems to the cubic NLS equation and the Hermite problem.

Application of Weierstrass Function

The paper discusses the properties of the Weierstrass p-function, emphasizing its role in generalizing trigonometric functions and expressing solutions to the KdV equation. The elliptic invariants $g_2$ and $g_3$ are defined, and special cases are presented. The author extends the Weierstrass p-function to $\mathbb{R}^3$ using a quaternionic version, aiming to model fluid flows with triply periodic structures. This quaternionic p-function is presented as a solution to quaternionic differential equations, comparable to the stationary version of the KdV equation.

Streamlines and Soliton Solutions

The streamlines of the flow are analyzed as eigenfunctions of a Hamiltonian. Madelung's fluid equation is used to derive traveling-wave solutions of the stationary KdV flow, coupled to related cubic NLS envelope solutions. The position vector of the vortex filament is shown to obey the envelope equation of the traveling waves, satisfying the NLS equation. The geodesics of these soliton surfaces are then represented using specific equations. The author connects these solutions to the Burgers vortex layer and Euler-Cornu spirals, explaining the von Kármán vortex street phenomenon as a diffraction pattern.

Benchmark Cases and Numerical Results

The paper presents two well-studied benchmark cases: the cylinder-wake problem and the driven-lid problem. For the cylinder-wake problem, the streamlines of the time-average velocity distribution exhibit tendril-like behavior. The geometric symmetries of the driven-lid cavity problem make it an underdetermined problem, preventing the direct application of the uniformizing Lemma 1. The paper discusses the incorporation of boundary conditions into the general solution using elliptic integral equations based on Lemma 1, which uniformizes curves of genus unity.

Conclusion

The paper concludes by asserting that the viscid incompressible Navier-Stokes equation reduces to physically coupled soliton systems described by the Burgers-Hopf and stationary KdV equations. This reduction leads to infinitely many equations of motion of the Calogero-Moser type. The author suggests that the initial conditions invoke a movable singularity within the domain of the solution, with the Error function and its derivatives serving as a soliton solution to the stationary KdV flow.