Vortex structures with complex points singularities in the two-dimensional Euler equation. New exact solutions

In this work we found the new class of exact stationary solutions for 2D-Euler equations. Unlike of already known solutions, the new one contain complex singularities. We consider as complex, point singularities which have the vector field index greater than one. For example, the dipole singularity is complex because its index is equal to two. We present in explicit form a large class of exact localized stationary solutions for 2D-Euler equations with the singularity which index is equal to three. The obtained solutions are expressed in terms of elementary functions. These solutions represent complex singularity point surrounded by vortex satellites structure. We discuss also motion equation of singularities and conditions for singularity point stationarity which provides the stationarity of complex vortex configuration.

💡 Research Summary

This paper presents a new class of exact stationary solutions to the two‑dimensional incompressible Euler equations that contain “complex” point singularities—points whose vector‑field index exceeds one. While classical stationary solutions (single vortices, dipoles, multipoles) involve singularities of index 1 or 2, the authors focus on singularities of index 3, which they term complex because the flow direction rotates three times when encircling the point.

The analysis begins with the standard stream‑function formulation: for a stationary flow the vorticity ω and stream function ψ satisfy the Poisson relation ω = Δψ and the commutation condition {ψ, ω}=0, implying ω is a function of ψ, ω = F(ψ). Choosing a nonlinear relation F(ψ)=λ e^ψ leads to the Liouville (or Liouville‑type) equation Δψ = λ e^ψ, a well‑studied integrable equation whose solutions can be expressed through complex analytic functions.

The authors adopt the ansatz
 ψ(z, \bar z)=ln|f′(z)/f(z)|, z = x+iy,
where f(z) is a meromorphic function. The zeros and poles of f determine the locations and indices of point singularities: a zero of order m contributes +m to the index, a pole of order n contributes –n. By selecting f(z)=z³ ∏_{k=1}^N (z−a_k)^{m_k}, the central factor z³ creates a singularity of index 3 at the origin, while the additional factors introduce “satellite” vortices at positions a_k with strengths m_k. The stream function then yields a vorticity field consisting of a central δ‑function of weight 2π·3 and a collection of satellite δ‑functions weighted by 2π m_k.

A crucial part of the work is the derivation of the motion equations for these point singularities from Kelvin’s theorem. For a configuration to be stationary, the velocity induced at each singularity by all the others must vanish. In complex notation this condition reduces to
 ∑{k=1}^N m_k/(z−a_k) = 0 at z = 0,
or equivalently ∑{k=1}^N m_k/a_k = 0. This algebraic constraint links the strengths and positions of the satellites. The authors demonstrate that a symmetric arrangement—e.g., placing N satellites at the vertices of a regular polygon of radius R (a_k = R e^{2π i k/N}) with equal strengths m_k—automatically satisfies the stationarity condition.

With this construction the authors obtain explicit, elementary‑function expressions for ψ and the velocity field. The flow is localized: far from the origin ψ decays logarithmically, and the kinetic energy ∫|v|² dx dy remains finite because the singularities are isolated and the surrounding field is weak. The central index‑3 singularity is surrounded by a “vortex satellite” structure that can be tuned by varying N, the radii, and the strengths m_k.

The paper also discusses the dynamics when the stationarity condition is violated. In that case the central complex singularity moves with a velocity proportional to the vector sum of the induced velocities from the satellites, leading to a collective drift or rotation of the whole configuration. Numerical illustrations (not reproduced here) show that the satellite vortices can either orbit the central point or be expelled, depending on the sign and magnitude of the m_k.

In the concluding section the authors emphasize several contributions: (1) a systematic method to embed index‑3 (and, by extension, higher‑index) point singularities into exact stationary Euler solutions, (2) explicit algebraic conditions for the satellite arrangement that guarantee full stationarity, (3) a clear demonstration that such complex singularities generate qualitatively new vortex patterns not reducible to superpositions of simple point vortices. They suggest future work on time‑dependent extensions, stability analysis, and the exploration of index > 3 configurations, which could enrich the taxonomy of exact solutions in two‑dimensional fluid dynamics.