Biharmonic Fields Research Articles

Some results on proper biharmonic vector field

In this paper, we establish necessary and sufficient conditions under which a vector field be biharmonic on the Riemannian manifold. We also construct some examples of proper biharmonic vector fields from a resolution of differential equations.

Planar biharmonic vector fields; potentials and traces

Planar biharmonic vector fields; potentials and traces

On the Biharmonicity of Vector Fields and Unit Vector Fields

Let (M, g) be a compact Riemannian manifold. Equipping its tangent bundle TM (resp. unit tangent bundle \(T_1M\)) with a pseudo-Riemannian g-natural metric G (resp. \({\tilde{G}}\)), we study the biharmonicity of vector fields (resp. unit vector fields) as maps \((M,g) \rightarrow (TM,G)\) (resp. \((M,g) \rightarrow (T_1M,{\tilde{G}})\)), as well as critical points of the bienergy functional restricted to the set \({\mathfrak {X}}(M)\) (resp. \({\mathfrak {X}}^1(M)\)) of vector fields (resp. unit tangent bundles) on M. Contrary to the Sasaki metric on TM, where the two notions are equivalent to the harmonicity of the vector field and then to its parallelism, we prove that for large classes of g-natural metrics on TM, the two notions are not equivalent. Furthermore, we give examples of vector fields which are biharmonic as critical points of the bienergy functional restricted to \({\mathfrak {X}}(M)\), but are not biharmonic maps. We provide equally examples of proper biharmonic vector fields (resp. unit vector fields), i.e. those which are critical points of the bienergy functional restricted to the set \({\mathfrak {X}}(M)\) (resp. \({\mathfrak {X}}^1(M)\)) without being harmonic.

New approach for optical electroostimistic phase with optical quasi potential energy

In this paper, we firstly review the optical circular motion with optical optimistic density in Heisenberg space. We define new quasi optical potential electromagnetic energy of quasi normal magnetic biharmonic fields. Moreover, we design the new characterization of uniformly optical electroosmotic phase and optical parallel transportation in cold plasma. Also, we collect new physical geometric designs for a optical optimistic density of normal magnetic biharmonic fields in Heisenberg space. Finally, we construct optical potential electromagnetic energy with respect to its electric field and sonic crystals.

Directed ratchet transport of cold atoms and fluxons driven by biharmonic fields: A unified view.

This paper discusses two retrodictions of the theory of ratchet universality which explain previous experimental results concerning directed ratchet transport of cold atoms in dissipative optical lattices in one case and of fluxons in uniform annular Josephson junctions in the other, both driven by biharmonic fields. It has to be emphasized that these retrodictions are in sharp contrast with the current standard explanation of such experimental results, and they offer optimal control of the ratchetlike motion of such entities. New experimental proposals with cold atoms and fluxons are discussed, providing additional tests for novel predictions from ratchet universality.

Quantum control operations with fuzzy evolution trajectories based on polyharmonic magnetic fields

We explore a class of quantum control operations based on a wide family of harmonic magnetic fields that vary softly in time. Depending on the magnetic field amplitudes taking part, these control operations can produce either squeezing or loop (orbit) effects, and even parametric resonances, on the canonical variables. For these purposes we focus our attention on the evolution of observables whose dynamical picture is ascribed to a quadratic Hamiltonian that depends explicitly on time. In the first part of this work we survey such operations in terms of biharmonic magnetic fields. The dynamical analysis is simplified using a stability diagram in the amplitude space, where the points of each region will characterise a specific control operation. We discuss how the evolution loop effects are formed by fuzzy (non-commutative) trajectories that can be closed or open, in the latter case, even hiding some features that can be used to manipulate the operational time. In the second part, we generalise the case of biharmonic fields and translate the discussion to the case of polyharmonic fields. Using elementary properties of the Toeplitz matrices, we can derive exact solutions of the problem in a symmetric evolution interval, leading to the temporal profile of those magnetic fields suitable to achieve specific control operations. Some of the resulting fuzzy orbits can be destroyed by the influence of external forces, while others simply remain stable.

Quantum vibrational resonance in a dual-frequency-driven Tietz-Hua quantum well.

We investigate the response of a quantum particle in the Tietz-Hua quantum potential driven by biharmonic fields: a low-frequency force and a very high frequency force. The response is characterized by the occurrence of a maximum in the first-order transition probability amplitude |s|^{2} under the influence of the applied fields. It is shown that in the absence of the high-frequency component of the applied fields, |s|^{2} shows a distinct sequence of resonances, whereas an increase in the amplitude of the high-frequency field induces minima in |s|^{2}. However, the |s|^{2} maximum occurs in the low-frequency regime where it may be considered otherwise weak in the presence of a single harmonic force.

The biharmonic homotopy problem for unit vector fields on 2-tori

The bienergy of smooth maps between Riemannian manifolds, when restricted to unit vector fields, yields two different variational problems depending on whether one takes the full functional or just the vertical contribution. Their critical points, called biharmonic unit vector fields and biharmonic unit sections, form different sets. Working with surfaces, we first obtain general characterizations of biharmonic unit vector fields and biharmonic unit sections under conformal change of the metric. In the case of a 2-dimensional torus, this leads to a proof that biharmonic unit sections are always harmonic and a general existence theorem, in each homotopy class, for biharmonic unit vector fields.

Biharmonic vector fields on pseudo-Riemannian manifolds

Abstract The bienergy of a vector field on a pseudo-Riemannian manifold ( M , g ) is defined to be the bienergy of the corresponding map ( M , g ) ↦ ( T M , g S ) , where the tangent bundle T M is equipped with the Sasaki metric g S . The constrained variational problem is studied, where variations are confined to vector fields, and the corresponding critical point condition characterizes biharmonic vector fields. We provide examples of biharmonic vector fields on three dimensional Lie groups equipped with a left-invariant Lorentzian metric and the Godel universe.

Biharmonic fields and mesh completion

We discuss bi-harmonic fields which approximate signed distance fields. We conclude that the bi-harmonic field approximation can be a powerful tool for mesh completion in general and complex cases. We present an adaptive, multigrid algorithm to extrapolate signed distance fields. By defining a volume mask in a closed region bounding the area that must be repaired, the algorithm computes a signed distance field in well-defined regions and uses it as an over-determined boundary condition constraint for the biharmonic field computation in the remaining regions. The algorithm operates locally, within an expanded bounding box of each hole, and therefore scales well with the number of holes in a single, complex model. We discuss this approximation in practical examples in the case of triangular meshes resulting from laser scan acquisitions which require massive hole repair. We conclude that the proposed algorithm is robust and general, and is able to deal with complex topological cases.

The biharmonicity of sections of the tangent bundle

The bienergy of a vector field on a Riemannian manifold \((M, g)\) is defined to be the bienergy of the corresponding map \((M, g)\mapsto (TM, g_{S})\), where the tangent bundle \(TM\) is equipped with the Sasaki metric \(g_{S}\). The constrained variational problem is studied, where variations are confined to vector fields, and the corresponding critical point condition characterizes biharmonic vector fields. Furthermore, we prove that if \((M, g)\) is a compact oriented \(m\)-dimensional Riemannian manifold and \(X\) a tangent vector of \(M\), then \(X\) is a biharmonic vector field of \((M, g)\) if and only if \(X\) is parallel. Finally, we give examples of non-parallel biharmonic vector fields in the case which the base manifold \((M, g)\) is non-compact.

Solving Wiener–Hopf problems without kernel factorization

A new approach to solving problems of Wiener–Hopf type is expounded by showing its implementation in two concrete and typical examples from fluid mechanics. The new method adapts mathematical ideas underlying the so-called unified transform method due to A. S. Fokas and collaborators in recent years. The method has the key advantage of avoiding what is usually the most challenging part of the usual Wiener–Hopf approach: the factorization of kernel functions into sectionally analytical functions. Two example boundary value problems, involving both harmonic and biharmonic fields, are solved in detail. The approach leads to fast and accurate schemes for evaluation of the solutions.

The bienergy of unit vector fields

The bienergy of a unit vector field on a Riemannian manifold \((M, g)\) is defined to be the bienergy of the corresponding map \((M, g)\mapsto (T_{1}M, g_{\mathrm{S}})\), where the unit tangent sphere bundle \(T_{1}M\) is equipped with the restriction of the Sasaki metric \(g_{\mathrm{S}}\). The constrained variational problem is studied, where variations are confined to unit vector fields, and the corresponding critical point condition characterizes biharmonic unit vector fields. Finally, we completely determine invariant biharmonic unit vector fields in three-dimensional unimodular Lie groups and the generalized Heisenberg groups \(H(1, r), r\ge 2\).

On the biharmonic vector fields

The problem studied in this paper is related to the biharmonicity of a vector field from a Riemannian manifold (M,g) to its tangent bundle TM equipped with the Sasaki metric g^s. We show that a vector field on a compact manifold is biharmonic if and only if is harmonic. We also investigate the biharmonicity of vector field of M, as a map from (M,g) to (TM,g^s).

Analysis of Stress on Interface Crack of Orthotropic and Isotropic Bi-Materials

This paper is concerned in semi-infinite interface crack of orthotropic and isotropic bi-materials and using the composite material fracture complex function method. By means of constructing special stress functions with two real singularity index and solving the problem of a class of generalized bi-harmonic equations , the stress and displacement fields of two dissimilar materials are obtained .Results demonstrate that the stress and displacement fields near the crack tip show mixed crack characteristics without oscillation.

Nonlinear properties of two-dimensional semiconductor superlattices in biharmonic fields

Conditions for the dynamic localization of electrons and electromagnetic transparency in a two-dimensional semiconductor superlattice in harmonic and biharmonic fields are obtained. The effect of static current excitation by a biharmonic field with orthogonally directed components is predicted.

Discrete soliton ratchets driven by biharmonic fields

Directed motion of topological solitons (kinks or antikinks) in the damped and ac-driven discrete sine-Gordon system is investigated. We show that if the driving field breaks certain time-space symmetries, the soliton can perform unidirectional motion. The phenomenon resembles the well known effects of ratchet transport and nonlinear harmonic mixing. Direction of the motion and its velocity depends on the shape of the ac drive. Necessary conditions for the occurrence of the effect are formulated. In comparison with the previously studied continuum case, the discrete case shows a number of new features: nonzero depinning threshold for the driving amplitude, locking to the rational fractions of the driving frequency, and diffusive ratchet motion in the case of weak intersite coupling.

Ratchetlike dynamics of fluxons in annular Josephson junctions driven by biharmonic microwave fields.

Experimental observation of the unidirectional motion of a topological soliton driven by a biharmonic ac force of zero mean is reported. The observation is made by measuring the current-voltage characteristics for a fluxon trapped in an annular Josephson junction that was placed into a microwave field. The dependence of the fluxon mean velocity at zero dc bias versus the phase shift between the first and second harmonic of the driving force is in qualitative agreement with theoretical expectations.

Nonlinear resonances in the absorption spectrum of the three-level atomic V system in strong polyharmonic and weak biharmonic fields

A three-level atomic system with a strong three-mode field and a probing biharmonic field at transitions 1→2 and 1⇆3 (1 is a common lower level), respectively, is studied theoretically by numerical simulation and an analysis of the mathematical expressions derived for the particular case of a symmetric arrangement of the strong field components relative to the transition frequency ω21. The absorption spectrum of the probing field components contains parametric supernarrow resonances against the background of resonances of the nonlinear interference effect. The ratios of the differences in the frequencies of the strong and probing fields, at which the supernarrow resonances appear, are found analytically. The results coincide with those obtained from numerical simulation.

Three-dimensional interference effects in the mechanical action of weak biharmonic fields upon particles with the J=0 → J=1 quantum transition

Explicit expressions are derived for the rectified radiative forces (RRFs) related to the action of a weak interfering optical field of an arbitrary three-dimensional (3D) configuration upon resonance particles featuring the J=0 → J=1 quantum transition. It is shown that, in contrast to the case of a monochromatic field, there are simple 3D biharmonic field configurations for which the ratio of the vortex and potential RRF components can be controlled by adjusting frequencies and polarizations of the interfering light waves. This modification of the RRF structure gives rise to qualitatively different types of both vortex and potential light-induced particle motions that may lead to a 3D spatial localization (confinement) of these particles within the cells of an effective optical lattice with a period significantly greater than the light wavelength. In particular, the particles may perform a stable rotational motion along closed trajectories inside the elementary cells.