Higher-order Derivative Terms Research Articles

Thermodynamic analysis of modified teleparallel gravity involving higher-order torsion derivative terms

The present study is elaborated to investigate the validity of thermodynamical laws in a modified teleparallel gravity based on higher-order derivative terms of torsion scalar. For this purpose, we consider spatially flat FRW model filled with perfect fluid matter contents. Firstly, we explore the possibility of existence of equilibrium as well as non-equilibrium picture of thermodynamics in this extended version of teleparallel gravity. Here, we present the first law and the generalized second law of thermodynamics (GSLT) using Hubble horizon. It is found that non-equilibrium description of thermodynamics exists in this theory with the presence of an extra term called as entropy production term. We also establish GSLT using the logarithmic corrected entropy. Further, by taking the equilibrium picture, we discuss validity of GSLT at Hubble horizon for two different F models. Using Gibbs law and the assumption that temperature of matter within Hubble horizon is similar to itself, We use different choices of scale factor to discuss the GSLT validity graphically in all scenarios. It is found that the GSLT is satisfied for a specified range of free parameters in all cases.

Weak field limit and gravitational waves in higher-order gravity

We derive the weak field limit for a gravitational Lagrangian density [Formula: see text], where higher-order derivative terms in the Ricci scalar [Formula: see text] are taken into account. The interest for this kind of effective theories comes out from the consideration of the infrared and ultraviolet behaviors of gravitational field and, in general, from the formulation of quantum field theory in curved spacetimes. Here, we obtain solutions in weak field regime both in vacuum and in the presence of matter and derive gravitational waves considering the contribution of [Formula: see text] terms. By using a suitable set of coefficients [Formula: see text], it is possible to find up to [Formula: see text] normal modes of oscillation with six polarization states with helicity 0 or 2. Here [Formula: see text] is the higher-order term in the [Formula: see text] operator appearing in the gravitational Lagrangian. More specifically: the mode [Formula: see text], with [Formula: see text], has transverse polarizations [Formula: see text] and [Formula: see text] with helicity 2; the [Formula: see text] modes [Formula: see text], with [Formula: see text], have transverse polarizations [Formula: see text] and non-transverse ones [Formula: see text], [Formula: see text], [Formula: see text] with helicity 0.

Fractional Model of Couple Stress Fluid for Generalized Couette Flow: A Comparative Analysis of Atangana–Baleanu and Caputo–Fabrizio Fractional Derivatives

Different from a Newtonian fluid, couple stress uid (CSF) includes a new material constant, which is responsible for couple stress and the lubricant viscosity. This material constant comes with the fourth-order spatial derivative term, and due to this higher-order derivative term in the momentum equation, this uid (CSF) is comparatively less investigated even for the classical uid problems. This paper aims to study the fractional model of CSF, based on the Atangana-Baleanu (AB) fractional derivatives de nition. Since this AB de nition is new, therefore, for the sake of comparison and correctness, this problem is also solved using the Caputo-Fabrizio (CF) fractional derivative de nition. The CSF is considered to flow between two parallel plates, one of which is at rest and the other is moving with constant velocity. The external pressure gradient is also applied. This type of ow situations is usually called generalized Couette ow. The problem is rst written in dimensionless form and then solved for the exact solution using the Laplace transform and the nite Fourier sine transform. The CSF velocity obtained via an AB fractional derivative is compared with the CSF velocity obtained via a CF fractional derivative approach, and the results obtained are shown graphically. The CSF results for interesting uid parameters are displayed in various graphs for both the AB and CF fractional derivatives. It is observed that the CSF velocities obtained with the AB and CF fractional derivatives are the same for unit time. For time less than one and greater than one, variation in CSF velocities is observed. In limiting sense, the present CSF solutions are reduced to a similar Newtonian uid problem solution in the absence of external pressure gradient.

Generalized quantum electrodynamics in the framework of generalized BRST transformation

Podolsky's electromagnetism which explains some of unresolved problems of usual QED has been investigated in the framework of finite field-dependent (FFBRST) BRST transformation. In this generalized QED (GQED), BRST invariant effective theories are written using generalized Lorenz gauge, and no mixing gauge condition , and it contains higher-order derivative terms. No mixing gauge is free from UV divergences in the radiative correction and easier to handle. We construct appropriate FFBRST transformation to obtain generating functional in no mixing gauge from that in Lorenz gauge. We show that all these BRST invariant effective theories in GQED are basically the same and are inter-connected with appropriately constructed FFBRST transformations.

Soliton solutions in two-dimensional Lorentz-violating higher derivative scalar theory

This paper shows a new approach to obtain analytical topological defects of a 2D Myers–Pospelov Lagrangian for two scalar fields. Such a Lagrangian presents higher-order kinetic terms, which lead us to equations of motion which are non-trivial to be integrated. Here we describe three possible scenarios for the equations of motion, named by timelike, spacelike and lightlike respectively. We started our investigation with a kink-like traveling wave Ansatz for the free theory, which led us to constraints for the dispersion relations of each scenario. We also introduced a procedure to obtain analytical solutions for the general theory in the three mentioned scenarios. We exemplified the procedure and discussed the behavior of the defect solutions carefully. It is remarkable that the methodology presented in this study led to analytical models, despite the complexity of the equations of motion derived from the 2D Myers–Pospelov Lagrangian. The methodology here tailored can be applied to several Lagrangians with higher-order derivative terms.

Local Discontinuous Galerkin Methods for the Two-Dimensional Camassa–Holm Equation

In this paper, the local discontinuous Galerkin method is developed to solve the two-dimensional Camassa–Holm equation in rectangular meshes. The idea of LDG methods is to suitably rewrite a higher-order partial differential equations into a first-order system, then apply the discontinuous Galerkin method to the system. A key ingredient for the success of such methods is the correct design of interface numerical fluxes. The energy stability for general solutions of the method is proved. Comparing with the Camassa–Holm equation in one-dimensional case, there are more auxiliary variables which are introduced to handle high-order derivative terms. The proof of the stability is more complicated. The resulting scheme is high-order accuracy and flexible for arbitrary h and p adaptivity. Different types of numerical simulations are provided to illustrate the accuracy and stability of the method.

Plausible black hole solutions in higher-derivative gravity

Here we obtain explicit black hole solutions in Extension Gravity models with high-order derivative terms, while the Lichnerowicz-type theorem simplifies our analysis by vanishing Ricci's scalar curvature. We find out two explicit static, spherical solutions that satisfy the presented action: the first one is the same usual Schwarzschild solution and the other one is the new non-Schwarzschild solution. It means that Schwarzschild's solution following the no-hair theorem can describe any black hole object on each gravity theory. Without considering the first law of thermodynamics for it, we show that the non-Schwarzschild solution is depending on its set of constants, and then we consider its entropy and other thermodynamic parameters for specific values of the constants. Dans le cadre de modèles de gravité étendue comportant des dérivées d'ordre supérieur, nous obtenons des solutions explicites représentant des trous noirs. Nous trouvons deux solutions sphériques statiques qui vérifient les équations : la première est la solution de Schwarzschild bien connue, alors que l'autre est une solution nouvelle non schwarzschildienne. Ainsi, la solution de Schwarzschild, qui résulte du théorème « sans cheveux », peut décrire un trou noir dans l'une ou l'autre des théories de la gravité. Sans partir de la première loi de la thermodynamique, nous montrons que la solution non schwarzschildienne dépend de certaines constantes, et nous évaluons son entropie et les autres grandeurs thermodynamiques pour des valeurs spécifiques des constantes.

Higher-order Skyrme hair of black holes

Higher-order derivative terms are considered as replacement for the Skyrme term in an Einstein-Skyrme-like model in order to pinpoint which properties are necessary for a black hole to possess stable static scalar hair. We find two new models able to support stable black hole hair in the limit of the Skyrme term being turned off. They contain 8 and 12 derivatives, respectively, and are roughly the Skyrme-term squared and the so-called BPS-Skyrme-term squared. In the twelfth-order model we find that the lower branches, which are normally unstable, become stable in the limit where the Skyrme term is turned off. We check this claim with a linear stability analysis. Finally, we find for a certain range of the gravitational coupling and horizon radius, that the twelfth-order model contains 4 solutions as opposed to 2. More surprisingly, the lowest part of the would-be unstable branch turns out to be the stable one of the 4 solutions.

Improved numerical AC superposition method for electron energy distribution functions

An improved AC superposition method is proposed to obtain the electron energy distribution functions (EEPFs) by reducing the effect of the high order derivative terms. In the AC superposition method, a small AC voltage is applied to the current-voltage curve, and the EEPF is obtained by the second harmonic current which is proportional to the second derivative of the current. If the AC voltage is not small compared to the electron temperature, the EEPF can be distorted by high derivative terms that are not ignorable. Thus, the proposed method uses the DC component and the second harmonics of the current to reduce the influence of the high order derivative terms. The fourth derivative term is removed and the coefficients of the higher order derivative terms become ignorable. The EEPF by this method shows that the resolution of the low energy part in the EEPFs is greatly improved.

Bright–dark solitary wave solutions of generalized higher-order nonlinear Schrödinger equation and its applications in optics

Analysis of short-pulse propagation in positive dispersion media for example in optical fibers and in shallow water, requires assorted high-order derivative terms. The different dynamical features underlying soliton interactions in the generalized higher-order nonlinear Schrödinger equation, which model multimode wave propagation under varied physical situations in nonlinear optics, are studied. In this paper, the new exact solitary solutions in generalized form of generalized higher-order nonlinear Schrödinger equation (NLSE) are constructed with the aid of symbolic computation by employing modified extended direct algebraic method. The complex physical phenomena of generalized higher-order NLSE can be understand from the obtained solutions. The computational work shows that the current method is simple, general, powerful, effective, and wider applicable. Moreover, several new complex higher-order NLSEs that arising in mathematical physics can also be solved by this efficient method.

Optimality conditions in set optimization employing higher-order radial derivatives

There are two approaches of defining the solutions of a set-valued optimization problem: vector criterion and set criterion. This note is devoted to higher-order optimality conditions using both criteria of solutions for a constrained set-valued optimization problem in terms of higher-order radial derivatives. In the case of vector criterion, some optimality conditions are derived for isolated (weak) minimizers. With set criterion, necessary and sufficient optimality conditions are established for minimal solutions relative to lower set-order relation.

Higher-order derivatives of Lyapunov functions and partial boundedness of solutions with partially controllable initial conditions

Certain sufficient criteria for the types of partial boundedness of solutions with partially controllable initial conditions are obtained in terms of higher-order derivatives of the Lyapunov functions.

Some results on sensitivity analysis in set-valued optimization

In the paper, the higher-order contingent derivative of a parametrized set-valued inclusion is first established. For its applications, we obtain sensitivity analysis of solution map in the decision variable space for a parametrized constrained set-valued optimization problem in terms of higher-order contingent derivatives.

Finite Element Method of BBM-Burgers Equation with Dissipative Term Based on Adaptive Moving Mesh

A finite element model is proposed for the Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation with a high-order dissipative term; the scheme is based on adaptive moving meshes. The model can be applied to the equations with spatial-time mixed derivatives and high-order derivative terms. In this scheme, new variables are needed to make the equation become a coupled system, and then the linear finite element method is used to discretize the spatial derivative and the fifth-order Radau IIA method is used to discretize the time derivative. The simulations of 1D and 2D BBM-Burgers equations with high-order dissipative terms are presented in numerical examples. The numerical results show that the method keeps a second-order convergence in space and provides a smaller error than that based on the fixed mesh, which demonstrates the effectiveness and feasibility of the finite element method based on the moving mesh. We also study the effect of the dissipative terms with different coefficients in the equation; by numerical simulations, we find that the dissipative termuxxplays a more important role thanuxxxxin dissipation.

Phase structure of theO(2)ghost model with higher-order gradient term

The phase structure and the infrared behaviour of the Euclidean 3-dimensional $O(2)$ symmetric ghost scalar field model with higher-order derivative term has been investigated in Wegner and Houghton's renormalization group framework. The symmetric phase in which no ghost condensation occurs and the phase with restored symmetry but with a transient presence of a ghost condensate have been identified. Finiteness of the correlation length at the phase boundary hints to a phase transition of first order. The results are compared with those for the ordinary $O(2)$ symmetric scalar field model.

Soliton, rational, and periodic solutions for the infinite hierarchy of defocusing nonlinear Schrödinger equations.

Analysis of short-pulse propagation in positive dispersion media, e.g., in optical fibers and in shallow water, requires assorted high-order derivative terms. We present an infinite-order "dark" hierarchy of equations, starting from the basic defocusing nonlinear Schrödinger equation. We present generalized soliton solutions, plane-wave solutions, and periodic solutions of all orders. We find that "even"-order equations in the set affect phase and "stretching factors" in the solutions, while "odd"-order equations affect the velocities. Hence odd-order equation solutions can be real functions, while even-order equation solutions are complex. There are various applications in optics and water waves.

A new and rigorous SPN theory for piecewise homogeneous regions

A new SPN theory valid for piecewise homogeneous regions is rigorously derived without mathematical approximations. The resulting SPN equations are the same as the conventional ones, but both the interface and the boundary conditions are different from the ad hoc conditions conventionally assumed. The new theory does not need the assumption of locally 1D behavior near a surface.Gelbard pointed out in 1960 that SPN is rigorously valid for an infinite domain 3D problem with constant total cross-section and arbitrarily distributed isotropic sources, and speculated that SPN could be a good approximation for practical applications. The Gelbard’s example does not involve any interface or boundary condition. For a generic problem with piecewise homogeneous regions, even though the SPN equations can still be rigorously derived for each homogeneous region, the interface or boundary condition poses the problem for generalizing the rigorous SPN theory. In this paper we point out that in each of the homogeneous regions, the solution to the SPN equations can be used to rigorously reconstruct an angular flux distribution which is a particular solution to the PN equations. The complete PN angular flux solution is the sum of this particular SPN angular flux solution and a set of additional functions that are solution to the homogeneous part of the PN equations. This particular SPN angular flux solution alone does not have enough independent functions to fulfill all the PN interface or boundary conditions. In this case, one can only satisfy a subset of the conditions. Because this subset of conditions cannot be decoupled from the full set of conditions, the whole core SPN solution so obtained cannot provide exactly the same scalar flux solution as that given by the PN equations. Nevertheless if this particular SPN angular flux solution is to be used alone, its corresponding interface and boundary conditions can be explicitly derived without introducing mathematical approximations. The resulting conditions are different from the conventionally used ad hoc conditions for SPN. The new interface and boundary conditions involve additional terms of higher order derivatives of the SPN solution and are consistent with what Selengut found for SP3 in 1970. The arguments and techniques needed to establish the above theory can in fact be all deduced from the detailed PN theory formulation for piecewise homogeneous regions that was worked out by Davison in 1957 and 1958.This new SPN theory is valid for the multi-group case as well and can be generalized to the case of anisotropic scattering. It may be a good candidate for practical applications to 3D pin-cell meshed whole core calculations, if the pin-cell meshes can be properly homogenized. The new interface and boundary conditions can be implemented in conventional SPN codes but require more work to revise the coupling equations, resulting in new SPN nodal equations. The discontinuity factors needed for using homogenized meshes can be generated without ambiguity by using the particular SPN angular flux solution as well. A physical interpretation for the SPN theory is suggested. It is believed that the missing steps are now filled to have a sound and complete SPN theory.

Polymer quantization, stability and higher-order time derivative terms

The possibility that fundamental discreteness implicit in a quantum gravity theory may act as a natural regulator for ultraviolet singularities arising in quantum field theory has been intensively studied. Here, along the same expectations, we investigate whether a nonstandard representation called polymer representation can smooth away the large amount of negative energy that afflicts the Hamiltonians of higher-order time derivative theories, rendering the theory unstable when interactions come into play. We focus on the fourth-order Pais–Uhlenbeck model which can be reexpressed as the sum of two decoupled harmonic oscillators one producing positive energy and the other negative energy. As expected, the Schrödinger quantization of such model leads to the stability problem or to negative norm states called ghosts. Within the framework of polymer quantization we show the existence of new regions where the Hamiltonian can be defined well bounded from below.

Black holes in D = 4 higher-derivative gravity

Extensions of Einstein gravity with higher-order derivative terms are natural generalizations of Einstein’s theory of gravity. They may arise in string theory and other effective theories, as well as being of interest in their own right. In this paper we study static black-hole solutions in the example of Einstein gravity with additional quadratic curvature terms in four dimensions. A Lichnerowicz-type theorem simplifies the analysis by establishing that they must have vanishing Ricci scalar curvature. By numerical methods we then demonstrate the existence of further black-hole solutions over and above the Schwarzschild solution. We discuss some of their thermodynamic properties, and show that they obey the first law of thermodynamics.

Four interpretations of temporal memory operators in the wave equation

Attenuation and dispersion in ultrasound and elastography may be modeled with convolution memory operators in the wave equation. When the attenuation follows an arbitrary frequency power law the memory is a time domain power law. Since that is also a fractional time derivative, a large body of literature on fractional derivatives then becomes available for analysis and simulation. It can also be shown that, e.g., an elementary grain shearing process in an unconsolidated saturated sediment results in fractional wave equations for both compressional and shear waves. Second, some of these wave equations can be derived from constitutive equations with memory operators, ensuring satisfaction of causality and the Kramers-Kronig relations. The fractional Kelvin-Voigt and Zener models and their resulting attenuation and dispersion will in particular be discussed. The fractional operators can also be interpreted as the result of an infinite number of basic processes. Therefore, third, a fractional wave equation can be shown to be the result of an infinite number of elementary relaxation processes. Fourth, the fractional constitutive equation can also be expressed as an infinite sum of integer order derivatives of higher order which is also equivalent to a wave equation with higher order derivative terms.