The class of surfaces in 3-space possessing nontrivial deformations which preserve principal directions and principal curvatures (or, equivalently, the shape operator) was investigated by Finikov and Gambier as far back as in 1933. We review some of the known examples and results, demonstrate the integrability of the c…
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48 results for “hydrodynamic shapes”
BézierGAN generates smooth curves from low-dimensional parameters.
problem Designing smooth curves for aerodynamic and hydrodynamic shapes.
method Generative model that maps low-dimensional latent representation to Bézier curve points.
result Generates diverse and realistic curves with consistent shape variation.
In this paper, we establish a fluid limit for a two--sided Markov order book model. Our main result states that in a certain asymptotic regime, a pair of measure-valued processes representing the "sell-side shape" and "buy-side shape" of an order book converges to a pair of deterministic measure-valued processes in a c…
Arnold discovered geodesics in fluid dynamics.
problem Understanding fluid motion through geometric perspectives.
method Exploring Euler's equations and their connection to geodesics on diffeomorphism manifolds.
result Geodesics in the space of volume-preserving diffeomorphisms correspond to solutions of Euler's equations.
Survey on matrix hydrodynamics, a 2D fluid model.
problem Modeling 2D incompressible fluids.
method Spatial discretization via quantization theory.
result Basic demonstration of matrix hydrodynamics.
Spray theory aids geometric hydrodynamics methods.
problem Geometric methods in hydrodynamics need a spray theory.
method Use convenient calculus to complete Michor and Kriegl's approach.
result A unitary approach to geometric hydrodynamics established.
Hydrodynamic structures linked to F-manifolds.
problem Hydrodynamic equations and their Hamiltonian structures.
method Introducing generalised (bi-)Hamiltonian structures and associating them with (bi-)flat F-manifolds.
result Generalised (bi-)Hamiltonian structures of hydrodynamic type can be associated with (bi-)flat F-manifolds.
In integrable hydrodynamic systems, coordinates exist where generators and symmetries are simple.
problem Existence of Riemannian invariants for integrable systems of hydrodynamic type.
method Finding coordinates where the generator and all symmetries are diagonal.
result In integrable hydrodynamic systems, there exist coordinates where the generator and all symmetries are diagonal.
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.
problem Establishing integrability for quasilinear systems.
method Study of algebraic geometry of characteristic correspondence and introduction of conjugate nets.
result Unified and extended results for three subclasses of quasilinear systems.
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.
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.
The paper studies geometric properties of hydrodynamical density manifolds.
problem Understanding the geometry of hydrodynamical density manifolds.
method Formulating connections, gradients, Hessians, parallel transports, and curvatures on these manifolds.
result Closed-form formulas for sectional curvatures in one-dimensional density manifolds.
Hydrodynamic hierarchy deformed using conservation laws.
problem Deforming a hydrodynamic hierarchy with non-vanishing Nijenhuis torsion.
method Using a chain of conservation laws to deform the hierarchy.
result The resulting hierarchy has non-vanishing Nijenhuis torsion but vanishing Haantjes tensor.
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…
We classify in this paper infinitesimal quasitrivial deformations of semisimple bihamiltonian structures of hydrodynamic type.
The goal of international trade theories is to explain the exchange of goods and services between different countries, aiming to benefit from it. Albeit the idea is very simple and known since ancient history, smart policy and business strategies need to be implemented by each subject, resulting in a complex as well as…
A Stochastic Cascade Model for FX Dynamicscond-mat.stat-mech
A time series model for the FX dynamics is presented which takes into account structural peculiarities of the market, namely its heterogeneity and an information flow from long to short time horizons. The model emerges from an analogy between FX dynamics and hydrodynamic turbulence. The heterogeneity of the market is m…
The paper proves conditions for Darboux integrability in diagonal hydrodynamic systems.
problem Conditions for Darboux integrability in diagonal hydrodynamic systems.
method Proof of conditions using Laplace transformation sequences and geometric interpretations.
result Diagonal systems of hydrodynamic type are Darboux integrable if and only if the corresponding systems for commuting flows are Darboux integrable.
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.
problem Geodesic equations on diffeomorphism groups.
method Proving smoothness of solutions and constructing exponential maps.
result Smooth solutions match boundary conditions.
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.
problem Open problems in fluid dynamics and invariant metrics.
method Variational settings, models for invariant metrics, Cauchy and boundary value problems.
result New constructions and recent developments in fluid dynamics.
Machine learning predicts ship performance changes over time.
problem Estimating ship hydrodynamic performance over time.
method Machine learning methods (NL-PCR, NL-PLSR, probabilistic ANN) calibrated with in-service data.
result Probabilistic ANN model performs best in predicting ship performance changes.
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.
Hydrodynamic Coupling of Particle Inclusions Embedded in Curved Lipid Bilayer Membranescond-mat.soft
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…
The generalised hodograph method for non-diagonalisable integrable systems of hydrodynamic typenlin.SI
Extends method for solving certain hydrodynamic systems.
problem Solving non-diagonalisable integrable systems of hydrodynamic type.
method Generalised hodograph method applied to F-manifolds with compatible connections.
result Provides general solution under certain assumptions.
Optimizes hydrokinetic turbine design using morphing and Bayesian optimization.
problem Designing optimal hydrokinetic turbine shapes due to high cost and geometric constraints.
method Design-by-Morphing (DbM) and Mixed variable, Multi-Objective Bayesian Optimization (MixMOBO).
result Optimized shapes lead to maximum power output with minimal evaluations.
Simplified Ricci curvature for spherical fluid dynamics models.
problem Studying stability in incompressible fluid dynamics on a sphere.
method Definition and calculation of Ricci curvature for two-dimensional hydrodynamics using finite-dimensional Zeitlin models.
result Strong numerical evidence suggests convergence of finite-dimensional approximations to infinite-dimensional limit, indicating average instability for high-frequency modes.
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…
New methods for solving hydrodynamic-type equations using quasi-rectifiable Lie algebras.
problem Solving systems of hydrodynamic-type equations.
method Introducing and analyzing quasi-rectifiable Lie algebras and vector fields.
result New methods for solving hydrodynamic-type equations.
The Madelung transform connects quantum mechanics and hydrodynamics.
problem Quantum mechanics and hydrodynamics equivalence for generic wave functions.
method Poisson geometry and coadjoint orbits of semidirect extensions of diffeomorphism groups.
result The Madelung transform provides a natural infinite-dimensional version of convexity results.
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.
problem Investigate the Euler equations and surface quasi-geostrophic equation family.
method Realize equations as geodesic equations on diffeomorphism groups and analyze Riemannian exponential maps.
result Show precise conditions for non-linear Fredholm maps of index 0.
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…
Introduces infinite-dimensional differential geometry using Bastiani calculus.
problem Calculus breakdown in infinite-dimensional settings.
method Uses Bastiani calculus for directional derivatives.
result Develops and connects infinite-dimensional Lie groups and weak Riemannian geometry.
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.
problem Pseudo-gauge ambiguity in relativistic hydrodynamics.
method Develops -contact geometry to describe thermodynamic states and conservation laws.
result Identifies pseudo-gauge freedom as the non-uniqueness of thermodynamic equilibrium redefinition.
Study identifies numerical signs of blow-up in hydrodynamic equations.
problem Determining if numerical results of blow-up are genuine or artifacts.
method Geometrically consistent spatiotemporal discretization of complexified Euler equations.
result Identification of a signature based on supremum norm growth rates of vorticity.
Geometric framework for Newton's equations on diffeomorphism groups.
problem Modeling fluid dynamics and related systems on geometric spaces.
method Geodesic approach and infinite-dimensional information geometry.
result Unified framework for various fluid dynamics equations.
Defines new bi-flat structures from integrable systems and flat coordinates.
problem Creating new bi-flat structures from integrable systems.
method Combining Frölicher-Nijenhuis bicomplex with Lauricella bi-flat structures.
result Defines multi-parameter families of Lauricella bi-flat structures.
New approach links 2D fluid dynamics to matrix theory.
problem Understanding swirling patterns in 2D fluids.
method Matrix hydrodynamics linking 2D fluid dynamics to matrix theory.
result Established connections between 2D hydrodynamics and matrix Lie theory.
This paper solves a complex equation using Lie symmetry approach to find solitary wave and multiple soliton solutions.
problem Solving a (3 + 1)-dimensional KdV type equation.
method Lie group of transformation method to find infinitesimal generators and commutator table.
result Exact solutions of KdV type equation in explicit form.
The nonlinear equations for the general nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type are derived and the integrability of these equations by the method of inverse scattering problem is proved. For these equations, the Lax pairs with a spectral parameter are presented. Moreover, we demo…
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…
Study semi-invariant metrics in hydrodynamics for better understanding of fluid motion.
problem Understanding fluid motion using semi-invariant Riemannian metrics.
method Geometric approach via geodesic initial value problem on diffeomorphism groups.
result Local and some global well-posedness results for geodesic initial value problem.
Hydrodynamics principles applied to finance, solving complex market models.
problem Understanding financial market dynamics using physics principles.
method Unified mathematical framework using Kelvin waves to solve differential and pseudo-differential equations.
result Solved various financial models including volatility and variance swaps.
We solve the problem of reducing to the simplest and convenient for our purposes, canonical form for an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature in order to get an effective construction of the integrable hierarchies related to all…