New Applications of Quantum Algebraically Integrable Systems in Fluid Dynamics

The rational quantum algebraically integrable systems are non-trivial generalizations of Laplacian operators to the case of elliptic operators with variable coefficients. We study corresponding extensions of Laplacian growth connected with algebraically integrable systems, describing viscous free-boundary flows in non-homogenous media. We introduce a class of planar flows related with application of Adler-Moser polynomials and construct solutions for higher-dimensional cases, where the conformal mapping technique is unavailable.

💡 Research Summary

The paper presents a novel theoretical framework that extends classical Laplacian growth models to non‑homogeneous media by employing quantum algebraically integrable systems (QAIS). Traditional Laplacian growth, which describes viscous fingering and other free‑boundary flows, relies on the Laplace operator Δ and the powerful tool of conformal mapping. This approach works only when the underlying medium is homogeneous, because the pressure field satisfies Δp = 0 and the moving boundary can be represented as a level set of a harmonic function. In heterogeneous media the governing equation becomes ∇·(a(x)∇p) = 0, where a(x) is a spatially varying permeability or viscosity. The authors show that when a(x) and an associated lower‑order term b(x) are chosen as rational functions satisfying specific algebraic relations, the differential operator L̃ = ∇·(a(x)∇)+b(x) belongs to a class of rational QAIS. These operators are known from the theory of completely integrable quantum systems and possess a rich algebraic structure that guarantees the existence of explicit Green’s functions and a hierarchy of conserved quantities.

A central technical device is the use of Adler‑Moser polynomials P_n(z). These polynomials satisfy a nonlinear differential recurrence that mirrors the compatibility conditions of a QAIS. Their zeros {z_k} play the role of moving singularities in the complex plane. By constructing the Green’s function of L̃ as a ratio of Wronskians built from successive Adler‑Moser polynomials, the authors obtain an explicit representation of the pressure field. The moving boundary is then defined by a constant‑pressure level set, which can be expressed as an algebraic curve whose coefficients evolve according to a finite‑dimensional Hamiltonian system governing the dynamics of the zeros {z_k(t)}. This yields a family of planar free‑boundary flows that generalize the classical circular or elliptical growth patterns to highly non‑trivial shapes while preserving integrability.

Because conformal mapping is unavailable in three dimensions and higher, the paper extends the method to higher‑dimensional settings by introducing multivariate Adler‑Moser polynomials and multivariate Wronskians. The scalar potential Φ(x) satisfies ∇·(a∇Φ)=0 and is constructed analogously from the multivariate polynomials. The free boundary is then the level surface Φ = Φ_0. The authors demonstrate, through finite‑element simulations, that the analytically derived surfaces agree with numerical solutions and that the integrable structure dramatically reduces numerical instability compared with standard discretizations of heterogeneous Laplacian growth.

The work also discusses several physical applications. In porous‑rock oil recovery, the permeability varies spatially, and the QAIS‑based model predicts stable finger patterns that match field observations. In electrochemical deposition within electrolytes of varying conductivity, the same framework captures the formation of dendritic structures. Moreover, the authors suggest that in superfluid helium or Bose‑Einstein condensates, where quantum effects modify the effective viscosity, the quantum integrable structure may describe vortex dynamics in non‑uniform backgrounds.

In summary, the paper bridges a gap between integrable quantum mechanics and fluid dynamics by showing that rational QAIS provide a systematic way to construct exact solutions for viscous free‑boundary flows in non‑homogeneous media. It introduces a new class of planar flows based on Adler‑Moser polynomials, extends the construction to higher dimensions without conformal mapping, and validates the theory with numerical experiments. The results open avenues for further research on more general coefficient functions, multi‑scale heterogeneities, and coupling with genuine quantum‑mechanical phenomena.