arXiv research

The paper studies geometric properties of hydrodynamical density manifolds.

The paper proves conditions for Darboux integrability in diagonal hydrodynamic systems.

We draw connections between the field of contact topology and the study of Beltrami fields in hydrodynamics on Riemannian manifolds in dimension three. We demonstrate an equivalence between Reeb fields (vector fields which preserve a transverse nowhere-integrable plane field) up to scaling and rotational Beltrami field…

Starting from a homogeneous polynomial in momenta of arbitrary order we extract multi-component hydrodynamic-type systems which describe 2-dimensional geodesic flows admitting the initial polynomial as integral. All these hydrodynamic-type systems are semi-Hamiltonian, thus implying that they are integrable according t…

Researchers identify surfaces with special fluid flow fields.

Study Godbillon-Vey invariants in non-Lorentzian spacetimes and fluid dynamics.

This article is a local analysis of integrable GL(2)-structures of degree 4. A GL(2)-structure of degree n corresponds to a distribution of rational normal cones over a manifold M of dimension (n+1). Integrability corresponds to the existence of many submanifolds that are spanned by lines in the cones. These GL(2)-stru…

Gaussian Process Hydrodynamics approximates fluid flow equations using probabilistic kernels.

We investigate the well-posedness of (i) the heat flow of harmonic maps from $R^n$ to a compact Riemannian manifold without boundary for initial data in BMO; and (ii) the hydrodynamic flow $(u,d)$ of nematic liquid crystals on $\mathbb R^n$ for initial data in $BMO^{-1}\times BMO$.

Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.

SPH-ParVI uses fluid dynamics to sample unknown densities efficiently.

In this paper we construct multiparametric families of two dimensional metrics with polynomial first integral. Such integrable geodesic flows are described by solutions of some semi-Hamiltonian hydrodynamic type system. We find infinitely many conservation laws and commuting flows for this system. This procedure allows…

We find necessary and sufficient conditions for a local geodesic flow of an affine connection on a surface to admit a linear first integral. The conditions are expressed in terms of two scalar invariants of differential orders 3 and 4 in the connection. We use this result to find explicit obstructions to the existence …

Hydrodynamics principles applied to finance, solving complex market models.

New approach links 2D fluid dynamics to matrix theory.

Tensor measures chirality for curves, even those with rough edges.

Survey on matrix hydrodynamics, a 2D fluid model.

We derive an analytic formula for the hydrodynamic Green function and the Robin function on every orientable surface admitting a hydrodynamic Killing vector field. Closed-form expressions are provided for all fourteen canonical Riemann surfaces, covering both compact and non-compact cases; the formulae satisfy the slip…

Hydrodynamic structures linked to F-manifolds.

In integrable hydrodynamic systems, coordinates exist where generators and symmetries are simple.

HyperCR Einstein--Weyl equations in 2+1 dimensions reduce to a pair of quasi-linear PDEs of hydrodynamic type. All solutions to this hydrodynamic system can be in principle constructed from a twistor correspondence, thus establishing the integrability. Simple examples of solutions including the hydrodynamic reductions …

Geometric approach links hydrodynamic integrability to compatible nets.

Survey and new results link hydrodynamics, molecular physics, and financial engineering.

New maximal families of compatible Poisson structures derived from geodesically equivalent metrics.

We investigate multi-dimensional Hamiltonian systems associated with constant Poisson brackets of hydrodynamic type. A complete list of two- and three-component integrable Hamiltonians is obtained. All our examples possess dispersionless Lax pairs and an infinity of hydrodynamic reductions.

A new complex space resolves projective structures on surfaces.

We prove that the bihamiltonian cohomology of a semisimple pencil of Poisson brackets of hydrodynamic type vanishes for almost all degrees. This implies the existence of a full dispersive deformation of a semisimple bihamiltonian structure of hydrodynamic type starting from any infinitesimal deformation.

Hydrodynamic hierarchy deformed using conservation laws.

We classify in this paper infinitesimal quasitrivial deformations of semisimple bihamiltonian structures of hydrodynamic type.

In the present work, the integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type are effectively constructed. For achieving this aim, first of all, the problem on the canonical form of a special type for compatible nonlocal Poisson brackets of hydrodynamic type is solv…

In this note we first set up an analogy between spin and vorticity of a perfect 2d-fluid flow, based on the Borel-Weil contruction of the irreducible unitary representations of SU(2), and looking at the Madelung-Bohm velocity attached to the ensuing spin wave functions. We also show that, in the framework of finite dim…

We prove in this paper the quasitriviality of a class of deformations of the one component bihamiltonian structures of hydrodynamic type.

Smooth solutions found for hydrodynamic equations.

We study bi-Hamiltonian systems of hydrodynamic type with non-singular (semisimple) non-local bi-Hamiltonian structures and prove that such systems of hydrodynamic type are diagonalizable. Moreover, we prove that for an arbitrary non-singular (semisimple) non-locally bi-Hamiltonian system of hydrodynamic type, there ex…

Geometric Hydrodynamics tackles open problems in fluid dynamics.

We prove that under certain linear reciprocal transformation, an evolutionary PDE of hydrodynamic type that admits a bihamiltonian structure is transformed to a system of the same type which is still bihamiltonian.

Machine learning predicts ship performance changes over time.

We present a brief spray theory necessary for the geometric method in hydrodynamics. We make use of the convenient calculus to complete a unitary approach started by P. Michor and A. Kriegl.

We develop theory and computational methods to investigate particle inclusions embedded within curved lipid bilayer membranes. We consider the case of spherical lipid vesicles where inclusion particles are coupled through (i) intramembrane hydrodynamics, (ii) traction stresses with the external and trapped solvent flui…

Extends method for solving certain hydrodynamic systems.

Simplified Ricci curvature for spherical fluid dynamics models.

MARL improves LBM stability and accuracy across scales.

New methods for solving hydrodynamic-type equations using quasi-rectifiable Lie algebras.

The Madelung transform connects quantum mechanics and hydrodynamics.

In this paper the well-known Dubrovin-Novikov problem posed as long ago as 1984 in connection with the Hamiltonian theory of systems of hydrodynamic type, namely, the classification problem for multidimensional Poisson brackets of hydrodynamic type, is solved. In contrast to the one-dimensional case, in the general cas…

Study the geometry of hydrodynamics equations using diffeomorphism groups.

We investigate the role of Hertling-Manin condition on the structure constants of an associative commutative algebra in the theory of integrable systems of hydrodynamic type. In such a framework we introduce the notion of F-manifold with compatible connection generalizing a structure introduced by Manin.

Geometrically describes pseudo-gauge freedom in relativistic hydrodynamics.