Hilbert Equation Research Articles

Propagation of spin waves in a lattice of laterally and vertically coupled YIG microwaveguides by changing the magnetization angle in linear and nonlinear modes

Purpose. Investigation of the joint manifestation of the effects of anisotropic signal propagation, coupling, and nonlinear power dependence of the medium parameters in a lattice of laterally and vertically coupled spin-wave (SW) microwaveguides. Consideration of the case of the influence of the rotation of the magnetization angle and the change of the lateral gap between microwaveguides located on the same substrate on the transverse profile of the spin-wave beam and the spatial localization of the SW amplitude. Methods. The method of micromagnetic modeling based on the numerical solution of the Landau–Lifshitz–Hilbert equation shows the possibility of controlling the direction of propagation of SW in an ensemble of laterally and vertically coupled iron yttrium garnet (YIG) microwaveguides by changing the magnetization angle. By the method of numerical integration of the system of coupled discrete nonlinear Schrodinger equations, the possibility of changing the transverse profile of the spin-wave beam by changing the level of the initial signal amplitude is shown. Results. The spatial distributions of the components of the dynamic magnetization of the SW excited in two microwaveguides located on the same substrate obtained in micromagnetic simulations indicate a change in the character of localization of the SW power in the output sections of the microwaveguides. At variation of the lattice magnetization angle, a shift of the threshold power value is observed, at which a characteristic curbing of the transverse width of the spin-wave beam in the nonlinear mode appears. Conclusion. When excitation of surface magnetostatic SW in a lattice of laterally and vertically coupled microwaveguides, a transformation of the transverse profile of the wave is observed at a deviation of the magnetization angle of the structure by 15º , which is manifested in the change of the SW length and its localization in each of the microwaveguides. The combined effects of dipole coupling, gyrotropy, and nonlinearity of the medium make it possible to control the value of the threshold power of the SW, at which the mode of diffractionless propagation of the spin-wave beam is realized in a single layer of the structure.

Highest Cusped Waves for the Burgers–Hilbert Equation

Highest Cusped Waves for the Burgers–Hilbert Equation

Full Field Solution for Remotely Loaded One Side Completely Debonded Short Rigid Line Inclusion Embedded in Soft Matrix: Two-Dimensional Analytical and Experimental Insights

Abstract The problem of a completely debonded short fiber (rigid line inclusion/anticrack) embedded in a 2D isotropic elastic soft matrix subjected to the remote loading condition is of fundamental interest. The current work investigates completely debonded anticrack embedded in a soft (isotropic) matrix using Kolosov Muskhelisvili's complex potential framework. Here two configurations are studied: debonded inclusion oriented (i) parallel and (ii) perpendicular to the loading direction. In particular, the potentials take the form of a non-homogeneous Riemann—Hilbert equation for the given problem. Upon solving analytical forms of potentials, the stress fields were obtained. The stress field for the fully debonded anticrack exhibited oscillatory singular behavior between r−3/4 and r−1/4 with the dependence on the oscillatory index ε and material constants. The correctness of the analytical solution was validated using numerical simulation and experiments based on the digital photoelasticity technique. The analytical results were in good agreement with the experimental and numerically obtained stress fields confirming the accuracy of it. The magnitude of singularity is quantified by defining a complex stress intensity factor at the tip of the discontinuity and compared with the experimentally estimated value. So far in the literature, no full-field analytical solution exists for the completely debonded rigid inclusion embedded in an isotropic soft matrix. The solution obtained in the present work is of fundamental importance in developing the constitutive properties of short fiber reinforced thermoplastic (SFRT) composites.

On the approximation of vorticity fronts by the Burgers–Hilbert equation

This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived.

The Wave Breaking for Whitham-Type Equations Revisited

We prove wave breaking (shock formation) for some Whitham-type equations which include the Burgers--Hilbert equation, the fractional Korteweg--de Vries equation, and the classical Whitham equation. The result seems to be new for the Burgers--Hilbert equation and has been proven independently in (Yang, SIAM J. Math. Anal., 53 (2021), pp. 5756--5802). In all cases we provide simpler proofs than the known ones.

Lipschitz Continuity of Solutions to Drift-Diffusion Equations in the Presence of Nonlocal Terms

We analyze the propagation of Lipschitz continuity of solutions to various linear and nonlinear drift-diffusion systems, with and without incompressibility constraints. Diffusion is assumed to be either fractional or classical. We derive conditions that guarantee the propagation of Lipschitz regularity by the incompressible NSE in the form of a non-local, one dimensional viscous Burgers-type inequality. We show the analogous inequality is always satisfied for the generalized viscous Burgers–Hilbert equation, in any spatial dimension, leading to global regularity. We also obtain a regularity criterion for the Navier–Stokes equation with fractional dissipation $$(-\Delta )^{\alpha }$$ , regardless of the power of the Laplacian $$\alpha \in (0,1]$$ , in terms of Hölder-type assumptions on the solution. If we drop the pressure term and analyze a linear drift-diffusion problem instead, we obtain regularity under a super-critical assumption on the drift when the power of diffusion is in (1/2, 1]. We also generalize previous known regularity results under subcritical assumptions on the drift when the power of diffusion is between (0, 1/2]. The technique we use builds upon the evolution of moduli of continuity as introduced by Kiselev, Nazarov, Volberg and Shterenberg.

On persistence properties in weighted spaces for solutions of the fractional Korteweg–de Vries equation

Persistence problems in weighted spaces have been studied for different dispersive models involving non-local operators. Generally, these models do not propagate polynomial weights of arbitrary magnitude, and the maximum decay rate is associated with the dispersive part of the equation. Altogether, this analysis is complemented by unique continuation principles that determine optimal spatial decay. This work is intended to establish the above questions for a weakly dispersive perturbation of the inviscid Burgers equation. More precisely, we consider the fractional Korteweg–de Vries equation, which comprises the Burgers–Hilbert equation and dispersive effects weaker than those of the Benjamin–Ono equation.

Stress analysis for an orthotropic elastic half plane with an oblique edge crack and stress intensity factors

The main purposes of the present paper are as follows: (1) to analytically derive the general solution (stress functions) for an orthotropic elastic half plane with a crack or a notch; (2) to derive the Riemann–Hilbert equation as the analytical method; (3) to solve the present problem using two methods, one is a Cauchy integral method and other is a Riemann–Hilbert method; and (4) to derive the general expressions of the stress intensity factor (SIF) for a crack problem. The stress functions obtained by the Riemann–Hilbert equation are compared with those obtained by Cauchy integral method. Then, it is confirmed that the same stress functions can be obtained. The stress functions are expressed by any elastic constants. Therefore, Mode I and II SIFs can be calculated for any elastic constants. Some examples are shown. It is stated from the results of the examples that the negative Mode I SIFs exist for certain oblique edge crack angles and elastic constants. Because the derived stress functions are expressed using a mapping function, other geometrical shapes can be analyzed by changing the mapping function.

Влияние амплитуды СВЧ-воздействия на спиновый ток границы платина/железоиттриевый гранат

A numerical study of the features of the propagation of spin waves in a waveguide made of yttrium iron garnet (YIG) and Fe-Rh alloy in the form of a plate, located on top of the central part of the YIG, is carried out. Based on the simulation results, the possibilities of controlling the dynamics of spin waves in the structure under study were also revealed. Micromagnetic numerical simulation was used to study the transfer of a spin-wave signal in a multimode mode by numerically solving the Landau – Lifshitz – Hilbert equation. Transformation of the transmission spectra of spin waves shows that the proposed structure will make it possible to control the propagation of spin-wave modes due to a sharp change in the Fe-Rh magnetization in the region of the magnetic phase transition temperature close to room temperature. In addition, the spin wave signal can be controlled by a small temperature change in the Fe-Rh plate generated by the laser radiation. The two-layer structure of YIG / Fe-Rh, from an applied point of view, can be used as a functional unit in planar magnon networks performing space-frequency demultiplexing and filtering of spin-wave modes.

Shock Formation of the Burgers--Hilbert Equation

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 17 February 2021Accepted: 08 June 2021Published online: 06 October 2021KeywordsBurgers--Hilbert equation, shock, self similarityAMS Subject Headings35Publication DataISSN (print): 0036-1410ISSN (online): 1095-7154Publisher: Society for Industrial and Applied MathematicsCODEN: sjmaah

Модовая фильтрация поверхностных магнитостатических волн в YIG/FeRh

A numerical study of the features of the propagation of spin waves in a waveguide made of yttrium-ron garnet (YIG) and Fe-Rh alloy in the form of a plate, located on top of the central part of the YIG, has been carried out. Based on the simulation results, the possibilities of controlling the dynamics of spin waves in the structure under study were also revealed. Micromagnetic numerical simulation was used to study the transfer of a spin-wave signal in a multimode mode by numerically solving the Landau – Lifshitz – Hilbert equation. Transformation of the transmission spectra of spin waves shows that the proposed structure will make it possible to control the propagation of spin-wave modes due to a sharp change in the Fe-Rh magnetization in the region of the magnetic phase transition temperature close to room temperature. In addition, the spin wave signal can be controlled by a small temperature change in the Fe-Rh plate generated by laser radiation. The two-layer structure of YIG/Fe-Rh, from an applied point of view, can be used as a functional unit in planar magnon networks performing space-frequency demultiplexing and filtering of spin-wave modes.

Stability and Uniqueness for Piecewise Smooth Solutions to a Nonlocal Scalar Conservation Law with Applications to Burgers--Hilbert Equation

In this paper, we show uniqueness and stability for the piecewise-smooth solutions to the Burgers--Hilbert equation constructed in Bressan and Zhang [Commun. Math. Sci., 15(1):165--184, 2017]. The Burgers--Hilbert equation is $u_t+(\frac{u^2}{2})_x=\mathbf{H}[u]$ where $\mathbf{H}$ is the Hilbert transform, a nonlocal operator. We show stability and uniqueness for solutions amongst a larger class than the uniqueness result in Bressan and Zhang. The solutions we consider are measurable and bounded, satisfy at least one entropy condition, and verify a strong trace condition. We do not have smallness assumptions. We use the relative entropy method and theory of shifts (see Vasseur [Handbook of Differential Equations: Evolutionary Equations, 4:323 -- 376, 2008]).

Singularity of subsonic and transonic crack propagations along interfaces of magnetoelectroelastic bimaterials

The unified method for addressing the propagation of subsonic and transonic cracks along the interfaces of anisotropic bimaterials in Shen and Nishioka (2000) is applied in this paper for the plain strain problem of subsonic and transonic crack propagation along the interfaces of anisotropic magnetoelectroelastic (MEE) bimaterials. Using a modified Eshelby–Stroh formalism and analytic continuation, the problem here leads to a Riemann–Hilbert problem. After solving the Riemann–Hilbert equation, near-tip asymptotic fields are obtained, which shows the properties of the subsonic and transonic crack propagation in an MEE bimaterial system. Finally, the numerical results related to different singularity powers of the crack-tip are presented and discussed for barium titanate-cobalt ferrite (BaTiO3CoFe2O4) composites. The effects of piezoelectricity and piezomagnetics are also examined.

Piecewise smooth solutions to the Burgers–Hilbert equation

The paper is concerned with the Burgers-Hilbert equation ut + (u /2)x = H[u], where the right hand side is a Hilbert transform. Unique entropy admissible solutions are constructed, locally in time, having a single shock. In a neighborhood of the shock curve, a detailed description of the solution is provided.

Interfacial line inclusion between two dissimilar thermo-electro-magneto-elastic solids

In this study, a rigid line inclusion on the interface between two dissimilar thermo-electro-magneto-elastic solids is considered. Based on the Muskhelishvili’s theory and extended Stroh formalism, it is reduced to the non-homogeneous Hilbert equation. According to the holomorphic continuation technology and Muskhelishvili’s theory, the exact solutions for both thermal and electro-magneto-elastic fields are derived. The explicit expressions of the multiple coupling fields are presented, and the generalized stress intensity factors are illustrated with the closed form. From the results, it can be observed the coupling of the thermo-electro-magneto-elastic effects. Moreover, the generalized stress has singularities and oscillatory properties near the inclusion tip.

Global Existence of Weak Solutions for the Burgers--Hilbert Equation

This paper establishes the global existence of weak solutions to the Burgers--Hilbert equation, for general initial data in ${\bf L}^2(\mathbb{R})$. For positive times, the solution lies in ${\bf L}^2\cap{\bf L}^\infty$. A partial uniqueness result is proved for spatially periodic solutions, as long as the total variation remains locally bounded.

Estimation of Heat and Mass Transfer Coefficients During Air-Freezing of Cucumber

Freezing of foods involves coupled heat and mass transfer. It is essential to optimally design the freezing equipment and maximise the efficiency of the freezing process. Therefore, it is necessary to estimate the heat and mass transfer coefficients. In this study, the surface heat and mass transfer coefficients were estimated employing the Hilbert equation as well as the corresponding thermal and diffusive boundary layers during freezing of unpeeled cucumber at low air temperatures (−10 to −18 and −25°C). The estimated mean heat transfer coefficients ranged from 10.99 W m−2 K−1 for 0.5 m s−1 up to 40.07 W m−2 K−1 for 5.0 m s−1. The respective mass transfer coefficients ranged from 8.98 up to 32.40 m s−1. The estimated heat transfer coefficients were compared with the respective ones calculated from Dincer and Dincer and Cengeli during air-cooling (22 to 2°C) of unpeeled cucumbers of similar size as well as other agricultural cylindrical products.

Enhanced Life Span of Smooth Solutions of a Burgers--Hilbert Equation

We consider an initial value problem for a quadratically nonlinear inviscid Burgers--Hilbert equation that models the motion of vorticity discontinuities. We use a normal form transformation, which is implemented by means of a near-identity coordinate change of the independent spatial variable, to prove the existence of small, smooth solutions over cubically nonlinear time-scales. For vorticity discontinuities, this result means that there is a cubically nonlinear time-scale before the onset of filamentation.

HIGH FREQUENCY SCATTERING BY A SECOND-ORDER GENERALIZED IMPEDANCE DISCONTINUITY ON A CYLINDRICALLY CURVED SURFACE

The aim of the present paper is to obtain explicit asymptotic expressions for the \transfer (difiraction) related to the difiraction of high frequency cylindrical waves from the discontinuity occurred in the material properties as well as in the thicknesses of a coated cylindrically curved metallic sheet characterized by the second order GIBCs. Relying on the locality of the high frequency difiraction phenomenon, the angular interval ' 2 (i…;…) is extended to the abstract inflnite space ' 2 (i1;1) wherein the difiracting structure is replaced by a two-part cylindrically curved second order impedance sheet ‰ = a extending from ' = i1 to ' = 1. The resulting boundary value problem is formulated as a Hilbert equation which is solved asymptotically in the high frequency limit. Some graphical results showing the efiects of various parameters on the transfer coe-cients are presented.

The Mixed Mode I and II Interface Crack in Piezoelectromagneto–Elastic Anisotropic Bimaterials

Taking the electric–magnetic field inside the interface crack into account, the interface crack problem of dissimilar piezoelectromagneto (PEMO)–elastic anisotropic bimaterials under in-plane deformation is investigated. The conditions to decouple the in-plane and anti-plane deformation is presented for PEMO–elastic biaterials with a symmetry plane. Using the extended Stroh’s dislocation theory of two-dimensional space and the analytic continuition principle of complex analysis, the interface crack problem is turned into a nonhomogeneous Hilbert equation in matrix notation. Four possible eigenvalues as well as four eigenvectors for the fundamental solution to the corresponding homogeneous Hilbert equation are found, so are four modes of singularities for the fields around the interface crack tip. These singularities are shown to have forms of r−(1∕2)±iϵ1 and r−(1∕2)±iϵ2, in which the bimaterial constants ϵ1 and ϵ2 are proven to be real numbers for practical dissimilar PEMO–elastic bimaterials. Compared with the solution for the interface crack of dissimilar elastic bimaterials without electro–magnetic properties, two new additional singularities are discovered for the interface crack in the PEMO–elastic bimaterial media. The electric–magnetic field inside the crack is solved by employing the “energy method,” which is based on finding the stationary point of the saddle surface of the energy release rate with respect to the electro–magnetic field inside the crack. Closed form expressions for the extended crack tip stress fields and crack open displacements are formulated, so are some other fracture characteristic parameters, such as the extended stress intensity factors and energy release rate (G) for dissimilar PEMO–elastic bimaterial solids. Finally, fundamental results and some conclusions are presented, which could have applications in the failure of piezoelectro/magneto–elastic devices.