Higher-order Derivative Terms Research Articles

Gauging the higher derivative field theories in Ostrogradsky formalism

The gaugings of the higher derivative fields theory are analyzed in Ostrogradsky formalism with the aid of BRST charge. Treating the terms of higher-order time derivatives of fields as independent dynamic variables, we are able to convert the higher derivative Lagrangian into the more suitable Ostrogradski Hamiltonian formalism. Using the knowledge of local BRST-cohomology, it is possible for us to work out the deformed functional equations of the BRST charge as well as the corresponding BRST-invariant Hamiltonian. Moreover, by inserting these deformed quantities into the original system and utilizing the canonical equations of motion of phase variables, we gain the usual local gauge symmetries of the resulting interacting theory directly through the standard Legendre transformations.

Fractional relaxation noises, motions and the fractional energy balance equation

Abstract. We consider the statistical properties of solutions of the stochastic fractional relaxation equation and its fractionally integrated extensions that are models for the Earth's energy balance. In these equations, the highest-order derivative term is fractional, and it models the energy storage processes that are scaling over a wide range. When driven stochastically, the system is a fractional Langevin equation (FLE) that has been considered in the context of random walks where it yields highly nonstationary behaviour. An important difference with the usual applications is that we instead consider the stationary solutions of the Weyl fractional relaxation equations whose domain is −∞ to t rather than 0 to t. An additional key difference is that, unlike the (usual) FLEs – where the highest-order term is of integer order and the fractional term represents a scaling damping – in the fractional relaxation equation, the fractional term is of the highest order. When its order is less than 1/2 (this is the main empirically relevant range), the solutions are noises (generalized functions) whose high-frequency limits are fractional Gaussian noises (fGn). In order to yield physical processes, they must be smoothed, and this is conveniently done by considering their integrals. Whereas the basic processes are (stationary) fractional relaxation noises (fRn), their integrals are (nonstationary) fractional relaxation motions (fRm) that generalize both fractional Brownian motion (fBm) as well as Ornstein–Uhlenbeck processes. Since these processes are Gaussian, their properties are determined by their second-order statistics; using Fourier and Laplace techniques, we analytically develop corresponding power series expansions for fRn and fRm and their fractionally integrated extensions needed to model energy storage processes. We show extensive analytic and numerical results on the autocorrelation functions, Haar fluctuations and spectra. We display sample realizations. Finally, we discuss the predictability of these processes which – due to long memories – is a past value problem, not an initial value problem (that is used for example in highly skillful monthly and seasonal temperature forecasts). We develop an analytic formula for the fRn forecast skills and compare it to fGn skill. The large-scale white noise and fGn limits are attained in a slow power law manner so that when the temporal resolution of the series is small compared to the relaxation time (of the order of a few years on the Earth), fRn and its extensions can mimic a long memory process with a range of exponents wider than possible with fGn or fBm. We discuss the implications for monthly, seasonal, and annual forecasts of the Earth's temperature as well as for projecting the temperature to 2050 and 2100.

Painlevé Test and Exact Solutions for (1 + 1)-Dimensional Generalized Broer–Kaup Equations

In this paper, the Painlevé integrable property of the (1 + 1)-dimensional generalized Broer–Kaup (gBK) equations is first proven. Then, the Bäcklund transformations for the gBK equations are derived by using the Painlevé truncation. Based on a special case of the derived Bäcklund transformations, the gBK equations are linearized into the heat conduction equation. Inspired by the derived Bäcklund transformations, the gBK equations are reduced into the Burgers equation. Starting from the linear heat conduction equation, two forms of N-soliton solutions and rational solutions with a singularity condition of the gBK equations are constructed. In addition, the rational solutions with two singularity conditions of the gBK equation are obtained by considering the non-uniqueness and generality of a resonance function embedded into the Painlevé test. In order to understand the nonlinear dynamic evolution dominated by the gBK equations, some of the obtained exact solutions, including one-soliton solutions, two-soliton solutions, three-soliton solutions, and two pairs of rational solutions, are shown by three-dimensional images. This paper shows that when the Painlevé test deals with the coupled nonlinear equations, the highest negative power of the coupled variables should be comprehensively considered in the leading term analysis rather than the formal balance between the highest-order derivative term and the highest-order nonlinear term.

Higher-Order Optimality Conditions in Set-Valued Optimization with Respect to General Preference Mappings

We present higher order necessary conditions for a model of welfare economics, where the preference mapping has a star-shape property. We assume that the preferences can be satiable and can be described by an arbitrary preference set, without the use of utility functions. These conditions are formulated in terms of higher-order directional derivatives of multivalued mappings, and the variable domination structure is not given by cones.

Almost second-order uniformly convergent numerical method for singularly perturbed convection–diffusion–reaction equations with delay

This paper deals with the numerical treatment of time-dependent singularly perturbed convection–diffusion–reaction equations with delay. In the considered equations, the highest order derivative term is multiplied by a perturbation parameter, ε taking arbitrary value in the interval . For small ε, the solution of the equations exhibits a boundary layer on the right side of the spatial domain. The considered equations contain a small delay on the convection and reaction terms of the spatial variable. The terms involving the delay are approximated using the Taylor series approximation. The resulting singularly perturbed parabolic convection–diffusion–reaction equations are treated using the Crank Nicolson method in time derivative discretization and the mid-point upwind finite difference method on piecewise uniform Shishkin mesh for the space variable derivative discretization. The stability and the uniform convergence of the scheme are investigated well. The proposed scheme gives almost first-order convergence in the spatial direction. The Richardson extrapolation technique is applied to accelerate the rate of convergence of the scheme in the spatial direction to order almost two. To validate the applicability of the scheme, numerical examples are presented and solved for different values of the perturbation parameter and delay parameter.

φenics: Vainshtein screening with the finite element method

Within the landscape of modified theories of gravity, progress in understanding the behaviour of, and developing tests for, screening mechanisms has been hindered by the complexity of the field equations involved, which are nonlinear in nature and characterised by a large hierarchy of scales. This is especially true of Vainshtein screening, where the fifth force is suppressed by high-order derivative terms which dominate within a radius much larger than the size of the source, known as the Vainshtein radius.=-1In this work, we present the numerical code φenics, building on the FEniCS library, to solve the full equations of motion from two theories of interest for screening: a model containing high-order derivative operators in the equation of motion and one characterised by nonlinear self-interactions in two coupled scalar fields. We also include functionalities that allow the computation of higher-order operators of the scalar fields in post-processing, enabling us to check that the profiles we find are consistent solutions within the effective field theory. These two examples illustrate the different challenges experienced when trying to simulate such theories numerically, and we show how these are addressed within this code. The examples in this paper assume spherical symmetry, but the techniques may be straightforwardly generalised to asymmetric configurations. This article therefore also provides a worked example of how the finite element method can be employed to solve the screened equations of motion. φenics is publicly available and can be adapted to solve other theories of screening.

Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation

This paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter ε , taking arbitrary value in the interval 0 , 1 . For small values of ε , solution of the problem exhibits an exponential boundary layer on the right side of the spatial domain. The properties and bounds of the solution and its derivatives are discussed. The considered singularly perturbed time delay problem is solved using the Crank-Nicolson method in temporal discretization and exponentially fitted operator finite difference method in spatial discretization. The stability of the scheme is investigated and analysed using comparison principle and solution bound. The uniform convergence of the scheme is discussed and proven. The formulated scheme converges uniformly with linear order of convergence. The theoretical analysis of the scheme is validated by considering numerical test examples for different values of ε .

Higher-Order Uniformly Convergent Numerical Scheme for Singularly Perturbed Differential Difference Equations with Mixed Small Shifts

This paper deals with numerical treatment of singularly perturbed differential difference equations involving mixed small shifts on the reaction terms. The highest-order derivative term in the equation is multiplied by a small perturbation parameter ε taking arbitrary values in the interval 0,1 . For small values of ε , the solution of the problem exhibits exponential boundary layer on the left or right side of the domain and the derivatives of the solution behave boundlessly large. The terms having the shifts are treated using Taylor’s series approximation. The resulting singularly perturbed boundary value problem is solved using exponentially fitted operator FDM. Uniform stability of the scheme is investigated and analysed using comparison principle and solution bound. The formulated scheme converges uniformly with linear order before Richardson extrapolation and quadratic order after Richardson extrapolation. The theoretical analysis of the scheme is validated using numerical test examples for different values of ε and mesh number N .

Fitted Numerical Scheme for Singularly Perturbed Differential Equations Having Two Small Delays

In this paper, singularly perturbed differential equations having delay on the convection and reaction terms are considered. The highest order derivative term in the equation is multiplied by a perturbation parameter epsilon taking arbitrary values in the interval (0; 1]. For small epsilon, the problem involves a boundary layer on the left or right side of the domain depending on the sign of the coefficient of the convective term. The terms involving the delay are approximated using Taylor series approximation. The resulting singularly perturbed boundary value problem is treated using exponentially fitted upwind finite difference method. The stability of the proposed scheme is analysed and investigated using maximum principle and barrier functions for solution bound. The formulated scheme converges independent of the perturbation parameter with rate of convergence O(N−1). Richardson extrapolation technique is applied to accelerate the rate of convergence of the scheme to order O(N−2). To validate the theoretical finding, three model examples having boundary layer behaviour are considered. The maximum absolute error and rate of convergence of the scheme are computed. The proposed scheme gives accurate and parameter uniformly convergent result.

New Travelling Wave Solution-Based New Riccati Equation for Solving KdV and Modified KdV Equations

Abstract A large family of explicit exact solutions to both Korteweg- de Vries and modified Korteweg- de Vries equations are determined by the implementation of the new extended direct algebraic method. The procedure starts by reducing both equations to related ODEs by compatible travelling wave transforms. The balance between the highest degree nonlinear and highest order derivative terms gives the degree of the finite series. Substitution of the assumed solution and some algebra results in a system of equations are found. The relation between the parameters is determined by solving this system. The solutions of travelling wave forms determined by the application of the approach are represented in explicit functions of some generalized trigonometric and hyperbolic functions and exponential function. Some more solutions with different characteristics are also found.

A hyperbolic reformulation of the Serre-Green-Naghdi model for general bottom topographies

We present a novel hyperbolic reformulation of the Serre-Green-Naghdi (SGN) model for the description of dispersive water waves. Contrarily to the classical Boussinesq-type models, it contains only first order derivatives, thus allowing to overcome the numerical difficulties arising from higher order derivative terms, especially in the context of high order discontinuous Galerkin finite element schemes. The proposed model reduces to the original SGN model when an artificial sound speed tends to infinity. Moreover, it is endowed with an extra conservation law from which the energy-type conservation law associated with the original SGN model is retrieved when the artificial sound speed goes to infinity. In order to provide a theoretical basis for the proposed model, a derivation from the vertical average of the compressible Euler equations has been proposed. The governing partial differential equations are then solved at the aid of high order ADER discontinuous Galerkin finite element schemes. The new model has been successfully validated against numerical and experimental results, for both flat and non-flat bottom. For bottom topographies with large variations, the new model proposed in this paper provides more accurate results with respect to the hyperbolic reformulation of the SGN model with the mild bottom approximation recently proposed in [1].

Curvilinear coordinate lattice Boltzmann simulation for necklace-ring beams in the nonlinear Schrödinger equation

Necklace-ring solitons have gained much attention due to their potential applications in optics and other scientific areas. In this paper, the numerical investigation of the nonlinear Schrödinger equation by using the curvilinear coordinate lattice Boltzmann method is proposed to study necklace-ring solitons. Different from those used in the general curvilinear coordinate lattice Boltzmann models, the lattices used in this work are uniform in two- and three-dimensional space. Furthermore, the model contains spatial evolution rather than time evolution to avoid the complexity of dealing with higher-order time derivative terms as well as to maintain the simplicity of the algorithm. Numerical experiments reproduce the evolution of two- and three-dimensional necklace-ring solitons. The truncation error analysis indicates that our model is equivalent to the Crank–Nicolson difference scheme.

A high-order FEM formulation for free and forced vibration analysis of a nonlocal nonlinear graded Timoshenko nanobeam based on the weak form quadrature element method

The purpose of this paper is to provide a high-order finite element method (FEM) formulation of nonlocal nonlinear nonlocal graded Timoshenko based on the weak form quadrature element method (WQEM). This formulation offers the advantages and flexibility of the FEM without its limiting low-order accuracy. The nanobeam theory accounts for the von Kármán geometric nonlinearity in addition to Eringen’s nonlocal constitutive models. For the sake of generality, a nonlinear foundation is included in the formulation. The proposed formulation generates high-order derivative terms that cannot be accounted for using regular first- or second-order interpolation functions. Hamilton’s principle is used to derive the variational statement which is discretized using WQEM. The results of a WQEM free vibration study are assessed using data obtained from a similar problem solved by the differential quadrature method (DQM). The study shows that WQEM can offer the same accuracy as DQM with a reduced computational cost. Currently the literature describes a small number of high-order numerical forced vibration problems, the majority of which are limited to DQM. To obtain forced vibration solutions using WQEM, the authors propose two different methods to obtain frequency response curves. The obtained results indicate that the frequency response curves generated by either method closely match their DQM counterparts obtained from the literature, and this is despite the low mesh density used for the WQEM systems.

Seventh and ninth orders characteristic-wise alternative WENO finite difference schemes for hyperbolic conservation laws

In this work, the characteristic-wise alternative formulation of the seventh and ninth orders conservative weighted essentially non-oscillatory (AWENO) finite difference schemes is derived. The polynomial reconstruction procedure is applied to the conservative variables rather than the flux function of the classical WENO scheme. The numerical flux contains a low order term and high order derivative terms. The low order term can use arbitrary monotone fluxes that can enhance the resolution and reduce numerical dissipation of the fine scale structures while capturing shocks essentially non-oscillatory. The high order derivative terms are approximated by the central finite difference schemes. The improved performance in terms of accuracy, essentially non-oscillatory shock capturing and resolution for the complex shocked flow with fine scale structures in the classical one- and two-dimensional problems is demonstrated. However, the inclusion of the high order derivative terms is prone to generate Gibbs oscillations around a strong discontinuity and might result in a negative density and/or pressure. Therefore, a positivity-preserving limiter [Hu et al. J. Comput. Phys. 242 (2013)] is adopted to ensure the positive density and pressure in the shocked flows with extreme conditions, such as Mach 2000 jet flow problem.

Galileon scalar electrodynamics

We construct a consistent model of Galileon scalar electrodynamics. The model satisfies three essential requirements: (1) The action contains higher-order derivative terms, and obey the Galilean symmetry, (2) Equations of motion also satisfy Galilean symmetry and contain only up to second-order derivative terms in the matter fields and, hence do not suffer from instability, and (3) local U(1) gauge invariance is preserved. We show that the non-minimal coupling terms in our model are different from that of the real scalar Galileon models; however, they match with the Galileon real scalar field action. We show that the model can lead to an accelerated expansion in the early Universe. We discuss the implications of the model for cosmological inflation.

Pure electromagnetic-gravitational interaction in Ho\u0159ava\u2013Lifshitz theory at the kinetic conformal point

We introduce the electromagnetic-gravitational coupling in the Hořava–Lifshitz framework, in 3+1 dimensions, by considering the Hořava–Lifshitz gravity theory in 4+1 dimensions at the kinetic conformal point and then performing a Kaluza–Klein reduction to 3+1 dimensions. The action of the theory is second order in time derivatives and the potential contains only higher order spacelike derivatives up to z=4, z being the critical exponent. These terms include also higher order derivative terms of the electromagnetic field. The propagating degrees of freedom of the theory are exactly the same as in the Einstein–Maxwell theory. We obtain the Hamiltonian, the field equations and show consistency of the constraint system. The conformal kinetic point is protected from quantum corrections by a second class constraint. At low energies the theory depends on two coupling constants, beta and alpha . We show that the anisotropic field equations for the gauge vector is a deviation of the covariant Maxwell equations by a term depending on beta -1. Consequently, for beta =1, Maxwell equations arise from the anisotropic theory at low energies. We also prove that the anisotropic electromagnetic-gravitational theory at the IR point beta =1, alpha =0, is exactly the Einstein–Maxwell theory in a gravitational gauge used in the ADM formulation of General Relativity.

Generalizations and applications of Young\u2019s integral inequality by higher order derivatives

In the paper, the authors generalize Young’s integral inequality via Taylor’s theorems in terms of higher order derivatives and their norms, andapply newly-established integral inequalities to estimate several concrete definite integrals, including a definite integral of a function which plays an indispensable role in differential geometry and has a connection with the Lah numbers in combinatorics, the exponential integral, and the logarithmic integral.

Universal lowest-twist in CFTs from holography

We probe the conformal block structure of a scalar four-point function in d ≥ 2 conformal field theories by including higher-order derivative terms in a bulk gravitational action. We consider a heavy-light four-point function as the boundary correlator at large central charge. Such a four-point function can be computed, on the gravity side, as a two-point function of the light operator in a black hole geometry created by the heavy operator. We consider analytically solving the corresponding scalar field equation in a near-boundary expansion and find that the multi-stress tensor conformal blocks are insensitive to the horizon boundary condition. The main result of this paper is that the lowest-twist operator product expansion (OPE) coefficients of the multi-stress tensor conformal blocks are universal: they are fixed by the dimension of the light operators and the ratio between the dimension of the heavy operator and the central charge CT. Neither supersymmetry nor unitary is assumed. Higher-twist coefficients, on the other hand, generally are not protected. A recursion relation allows us to efficiently compute universal lowest-twist coefficients. The universality result hints at the potential existence of a higher-dimensional Virasoro-like symmetry near the lightcone. While we largely focus on the planar black hole limit in this paper, we include some preliminary analysis of the spherical black hole case in an appendix.

Seidel aberration coefficients: an alternative computational method.

An alternative method to compute Seidel aberrations is presented that utilizes real pseudoparaxial skew rays traveling through a rotationally-symmetric lens system as a Taylor series expansion in terms of object height and ray-direction spherical coordinates. Expressions for defocus, lateral magnification, and Seidel primary ray aberration coefficients are obtained in terms of numerically-determined higher-order partial derivatives of rays which are proximate to the optical axis. In contrast to commonly used methods, the new form of the aberration coefficients is related to unit entrance pupil radius and unit object height. The proposed methodology can be extended to derive higher-order ray aberration coefficients.

An ostentatious model of cosmological scalar–tensor theory

We consider a novel model of gravity with a scalar field described by the Lagrangian with higher-order derivative terms in a cosmological context. The model has the same solution for the homogeneous and isotropic universe as in the model with the Einstein gravity, notwithstanding the additional higher-order terms. Finally, a possible modification scenario is briefly discussed.