Higher-order Derivative Terms Research Articles

The exact solutions to the generalized (2+1)-dimensional nonlinear wave equation

Due to the importance of the nonlinear partial differential equations in applied physics and engineering, many mathematicians and physicists are interesting to the nonlinear partial differential equations. One of the main tasks of studying the nonlinear partial differential equations is to obtain the exact solutions for the nonlinear partial differential equations. In this paper, we study the exact solutions of a generalized (2+1)-dimensional nonlinear partial differential equation. According to the modified hyperbolic function method and the traveling transformation method, the generalized (2+1)-dimensional nonlinear partial differential equation is changed into an ordinary differential equation. By taking the homogeneous balance between the highest order nonlinear terms and the highest order derivative terms, we change the differential equation into an algebraic system. By solving the algebraic system, we obtain the solutions for the nonlinear partial differential equation. At last, some solutions and their figures are given by specific parameters. All the solutions conclude the trigonometric function solutions, the positive hyperbolic function solutions, and the hyperbolic trigonometric function solutions. These solutions exhibit the dynamics of nonlinear waves and the solutions are useful for the study on interaction behavior of nonlinear waves in shallow water, plasma, nonlinear optics and Bose–Einstein condensates.

A decoupled, linearly implicit and high-order structure-preserving scheme for Euler–Poincaré equations

It is challenging to develop high-order structure-preserving finite difference schemes for the modified two-component Euler–Poincaré equations due to the nonlinear terms and high-order derivative terms. To overcome the difficulties, we introduce a bi-variate function and carefully choose the intermediate average variable in the temporal discretization. Then, we obtain a decoupled and linearly implicit scheme. It is shown that the fully-discrete scheme can keep both the discrete mass and energy conserved. And the fully-discrete scheme has fourth-order accuracy in the spatial direction and second-order accuracy in the temporal direction. Several numerical examples are given to confirm the theoretical results.

Comparison and application of downward continuation methods for potential field data

Downward continuation highlights the details of the gravity and magnetic anomalies and improves the horizontal resolution of potential field data. However, downward continuation acts as a high-pass filter. As depth increases, any high-frequency noise will be quickly amplified and mask the effective information in the continuation field. We review 13 popular downward continuation methods and categorize them into five classes. The methods are compared and discussed in terms of continued depth and noise robustness using synthetic data and then are applied to field data from the Pobei area in Xinjiang Province, China. It is found that the noise robustness of the integral equation approach is improved using the iterative Landweber form. The frequency-domain methods based on high-frequency truncation do not require iterations and have a fast calculation speed. Using the Laplacian of the Gaussian operator to calculate the second-order vertical derivative enhances the noise robustness of the Taylor series expansion-based techniques. Segmentation of the continued depth followed by downward continuation increases the accuracy of extrapolation methods. High-order derivative terms located in the denominator of the expansion series cause instability in the continuous fractional series expansion-based techniques, and the obtained results have to be further proceeded with median filtering.

TaBE: Decoupling spatial and spectral processing with Taylor’s unfolding method in the beamspace domain for multi-channel speech enhancement

In recent years, significant advancements have been made in neural beamforming, leveraging spectral and spatial cues to enhance their performance in multi-channel speech enhancement. When the frame-wise processing mechanism is required, there exists a trade-off between performance and algorithmic delay for existing all-neural beamformers. However, from the perspective of multi-source information fusion, the network is often encapsulated into a black box to entangle and fuse the spatial and spectral features into a non-linear feature space, which hinders our understanding of how they work collaboratively for target speech extraction. In this regard, this paper proposes to decouple the spatial and spectral domain processing inspired by Taylor’s approximation theory. Specifically, we reformulate the time-variant beamforming defined in the spatial domain into the adaptive weighting and mixing of different beam components in the beamspace domain. This reformulation enables us to model the recovery of target speech as a weighted sum operation in the beamspace domain, where each beam component is associated with an introduced unknown term for residual interference cancellation. By virtue of Taylor’s series expansion, the recovery process can be decomposed into the superimposition of the 0th-order non-derivative and high-order derivative terms, where the former acts as spatial filtering in the spatial domain, and the latter serves as a residual interference canceller in the spectral domain. We conduct extensive experiments on the spatialized LibriSpeech and L3DAS Challenge datasets. Experimental results show that, compared with existing advanced approaches, the proposed method not only achieves competitive performance in terms of multiple objective metrics but also provides feasible guidance in multi-channel speech enhancement pipeline design.

On the well-posed variational principle in degenerate point particle systems using embeddings of the symplectic manifold

Abstract A methodology on making the variational principle well-posed in degenerate systems is constructed. In the systems including higher-order time derivative terms being compatible with Newtonian dynamics, we show that a set of position variables of a coordinate system of a given system has to be fixed on the boundaries and that such systems are always Ostrogradski stable. For these systems, Frobenius integrability conditions are derived in explicit form. Relationships between integral constants indicated from the conditions and boundary conditions in a given coordinate system are also investigated by introducing three fundamental correspondences between Lagrange and Hamilton formulation. Based on these ingredients, we formulate problems that have to be resolved to realize the well-posedness in the degenerate systems. To resolve the problems, we compose a set of embeddings that extract a subspace holding the symplectic structure of the entire phase space in which the variational principle should be well-posed. Using these embeddings, we establish a methodology to set appropriate boundary conditions that the well-posed variational principle demands. Finally, we apply the methodology to examples and summarize this work as a three-step procedure such that one can use just by following it.

New black hole solutions in three-dimensional f(R) gravity

We construct two new classes of analytical solutions in three-dimensional spacetime in the framework of f(R) gravity whose field equations did not have cosmological constant. These solutions cannot coincide with Einstein’s general relativity due to the non-allowable vanishing of the dimensional parameter that characterizes the higher-order curvature. The first class of this solutions represents a non-rotating black hole (BH) while the second class corresponds to a rotating BH solution. The Ricci scalar of these BH solutions have non-trivial values and are described by the gravitational mass M, two angular momenta J and J1, and an effective cosmological constant Λeff that mainly comes from the contribution of the higher-order curvature terms. Interestingly enough, we observe that in contrast to Bañados–Teitelboim–Zanelli (BTZ) solution, which has only causal singularities and its scalar invariants are constant everywhere, the scalar invariants of our solutions indicate strong singularities for spacetime. This implies that the higher order derivative terms that come from f(R) gravity changes the casual structure of the spacetime. Indeed, the scalar invariants of the solutions behave as O1rn where n>0. The behavior of the invariants as r→0 become strong as compared with the behavior of black holes that reproduce from GR. Furthermore, we construct the forms, rotating/non-rotating, of the f(R) function showing that they behave as polynomial functions. Finally, we show that the obtained solutions are stable because of the positivity of their heat capacity, and also from the condition of Ostrogradski which states that the second derivative of f(R) should have a positive value.

An efficient computational technique for higher order KdV equation arising in shallow water waves

This paper focuses on the computational study of seventh-order KdV equations via the quintic B-spline (QBS) finite element method (FEM). The KdV equations contain higher-order derivative terms. We transform the higher order into lower order and obtain a coupled system to ease the solution procedure. Forward difference combined with the theta-weighted scheme is used to discretize the temporal part. In addition, the quintic B-spline approximates the functions and their derivatives. The Von Neumann method is used to check stability of the scheme. The performance of the proposed method is evaluated using four test problems. The numerical results are presented in the tables to evaluate the accuracy and proficiency of the scheme at various collocation points. Furthermore, the approximate and exact solutions are compared graphically which match each other, showing validation of the proposed method.

Reciprocal Bäcklund transformations and travelling wave structures of some nonlinear pseudo-parabolic equations

The equations in which time derivative appears in the highest order derivative term are called pseudo-parabolic equations. In this article, we will propose new singular and dark soliton solutions to a few such equations. We first find the symmetry algebra of such equations, and then by invariant reduction we obtain the exact solutions. Then, we classify the solutions by using their Lie subalgebras. Also, we will discuss the travelling wave solutions by using tanh–coth method. Moreover, we will discuss reciprocal Bäcklund transformations of the conservation laws.

Isogeometric dual reciprocity BEM for solving non-Fourier transient heat transfer problems in FGMs with uncertainty analysis

In this paper, the precise integration multi-patch isogeometric dual reciprocity boundary element method (IG-DRBEM) of the non-Fourier transient heat transfer problem in functionally graded materials (FGMs) is proposed, and the stochastic analysis of multidimensional material uncertainty by hyperbolic truncated adaptive sparse polynomial chaos expansion (PCE) is established based on the determine theory. Up to now, most research results of the isogeometric analysis boundary element method (IGABEM) in transient heat transfer problems are based on Fourier's law. The main reason is the higher-order derivative term with respect to time in the non-Fourier heat transfer control equation leads to additional domain integrals in the boundary-domain integral equation. The dual reciprocity method (DRM) can transform the domain integral to the boundary conveniently and maintain the dimension reduction advantage of the BEM. The introduction of the precise integration method (PIM) can improve the stability and accuracy of the time-domain solution. In the random analysis based on IG-DRBEM, a sparse PCE strategy with hyperbolic truncation is constructed to overcome the dimensional disaster difficulty of traditional PCE. Through the proposed adaptive iterative process, a relatively small number of PCE terms are retained compared with the full PCE on the premise of meeting the error estimation requirements. The presented method is verified to have good accuracy and high efficiency by the complex model similar to actual engineering, which further expands the application extent of IG-DRBEM.

Brownian axionlike particles

We study the non-equilibrium dynamics of a pseudoscalar axion-like particle (ALP) weakly coupled to degrees of freedom in thermal equilibrium by obtaining its reduced density matrix. Its time evolution is determined by the in-in effective action which we obtain to leading order in the (ALP) coupling but to \emph{all orders} in the couplings of the bath to other fields within or beyond the standard model. The effective equation of motion for the (ALP) is a Langevin equation with noise and friction kernels obeying the fluctuation dissipation relation. A ``misaligned'' initial condition yields damped coherent oscillations, however, the (ALP) population increases towards thermalization with the bath. As a result, the energy density features a mixture of a cold component from misalignment and a hot component from thermalization with proportions that vary in time $(cold)\,e^{-\Gamma t}+(hot)\,(1-e^{-\Gamma t})$, providing a scenario wherein the ``warmth'' of the dark matter evolves in time from colder to hotter. As a specific example we consider the (ALP)-photon coupling $g a \vec{E}\cdot \vec{B}$ to lowest order, valid from recombination onwards. For $T \gg m_a$ the long-wavelength relaxation rate is substantially enhanced $\Gamma_T = \frac{g^2\,m^2_a\,T}{16\pi} $. The ultraviolet divergences of the (ALP) self-energy require higher order derivative terms in the effective action. We find that at high temperature, the finite temperature effective mass of the (ALP) is $m^2_a(T) = m^2_a(0)\Big[ 1-(T/T_c)^4\Big]$, with $T_c \propto \sqrt{m_a(0)/g}$, \emph{suggesting} the possibility of an inverted phase transition, which when combined with higher derivatives may possibly indicate exotic new phases. We discuss possible cosmological consequences on structure formation, the effective number of relativistic species and birefringence of the cosmic microwave background.

Can we bypass no-go theorem for Ricci-inverse gravity?

Recently, Amendola et al. proposed a geometrical theory of gravity containing higher-order derivative terms. The authors introduced anticurvature scalar $(A)$, which is the trace of the inverse of the Ricci tensor ($A^{\mu\nu} = R_{\mu\nu}^{-1}$). In this work, we consider two classes of Ricci-inverse -- Class I and Class II -- models. Class I models are of the form $f(R, A)$ where $f$ is a function of Ricci and anticurvature scalars. Class II models are of the form ${\cal F}(R, A^{\mu\nu}A_{\mu\nu})$ where ${\cal F}$ is a function of Ricci scalar and square of anticurvature tensor. For both these classes of models, we numerically solve the modified Friedmann equations in the redshift range $1500 < z < 0$. We show that the late-time evolution of the Universe, i.e., evolution from matter-dominated epoch to accelerated expansion epoch, \emph{can not} be explained by these two classes of models. Using the reduced action approach, we show that we \emph{can not bypass} the no-go theorem for Ricci-inverse gravity models. Finally, we discuss the implications of our analysis for the early-Universe cosmology.

Reconstruction of cosmological models in a general gravitational framework involving torsion scalar

This study is elaborated to reconstruct some cosmological models in the gravitational framework of an extended teleparallel gravity based on the higher-order derivative terms of torsion scalar. To achieve this goal, we consider spatially flat FRW model with perfect fluid as background matter distribution. In this framework, we find some interesting exact solutions by using the hybrid and intermediate forms of scale factor via a well-known reconstruction scheme. For exploring physical significance of the reconstructed solutions, we analyze the behavior of dark energy EoS parameter, the validity of energy condition bounds as well as the generalized second law of thermodynamics graphically. For investigating the stability of reconstructed solutions, we check the behavior of speed of sound [Formula: see text] in each case. It is seen that in de Sitter case, the reconstruction of full Lagrangian function is not possible as it reduces to the case of simple [Formula: see text] gravity which has already been discussed in literature. While for power law form of scale factor, it is found that reconstruction is possible and most of the obtained solutions favor the quintessence era of cosmic evolution. Further, it is concluded that both the energy conditions and generalized second thermodynamical law hold for the reconstructed solutions and these solutions exhibit stable behavior (as [Formula: see text]). Lastly, in case of intermediate form of scale factor, the resulting differential equations are very complicated and hence the analytical solution is possible only for the simplest case which, further, indicates cosmologically promising behavior.

Convergence analysis for forward and inverse problems in singularly perturbed time-dependent reaction-advection-diffusion equations

In this paper, by employing the asymptotic expansion method, we prove the existence and uniqueness of a smoothing solution for a time-dependent nonlinear singularly perturbed partial differential equation (PDE) with a small-scale parameter. As a by-product, we obtain an approximate smooth solution, constructed from a sequence of reduced stationary PDEs with vanished high-order derivative terms. We prove that the accuracy of the constructed approximate solution can be in any order of this small-scale parameter in the whole domain, except a negligible transition layer. Furthermore, based on a simpler link equation between this approximate solution and the source function, we propose an efficient algorithm, called the asymptotic expansion regularization (AER), for solving nonlinear inverse source problems governed by the original PDE. The convergence-rate results of AER are proven, and the a posteriori error estimation of AER is also studied under some a priori assumptions of source functions. Various numerical examples are provided to demonstrate the efficiency of our new approach.

Semi-Lagrangian forward methods for some time-dependent nonlinear partial differential equations

In this article, we study one-step Semi-Lagrangian forward method for computing the numerical solutions of time-dependent nonlinear partial differential equations with initial and boundary conditions in one space dimension. Comparing with classic Semi-Lagrangian method, this method is more straight forward to analyze and implement. This method is based on Lagrangian trajectory from the departure points (regular nodes) to the arrival points by Runge-Kutta methods. The arrival points are traced forward from the departure points along the trajectory of the path. Most likely the arrival points are not on the regular grid nodes. However, it is convenient to approximate the high order derivative terms in spatial dimension on regular nodes. The convergence and stability are studied for the explicit methods. The numerical examples show that those methods work very efficient for the time-dependent nonlinear partial differential equations. For more information see https://ejde.math.txstate.edu/conf-proc/26/g1/abstr.html

Vlasov-Poisson equation in weighted Sobolev space \(W^{m, p}(w)\)

In this paper, we are concerned about the well-posedness of Vlasov-Poisson equation near vaccum in weighted Sobolev space \(W^{m, p}(w)\). The most difficult part comes from estimates of the electronic term \(\nabla_{x}\phi\). To overcome this difficulty, we establish the \(L^p\)-\(L^q\) estimates of the electronic term \(\nabla_{x}\phi\); some weight is introduced as well to obtain the off-diagonal estimate. The weight is also useful when it comes to control the higher-order derivative term.

Approximating compact objects in bootstrapped Newtonian gravity: use of the canonical potential

We consider compact objects in a classical and non-relativistic generalisation of Newtonian gravity, dubbed bootstrapped Newtonian theory, which includes higher-order derivative interaction terms of the kind generically present in the strong-field regime of gravity. By means of a field redefinition, the original bootstrapped Newtonian action is written in a canonical Newtonian form with non-linear source terms. Exact analytic solutions remain unattainable, but we show that perturbative solutions of the canonical theory can be efficiently used to derive approximate descriptions of compact objects. In particular, using the canonical potential, we can more directly and generally show that the Arnowitt–Deser–Misner mass differs from the (Newtonian) proper mass due to the non-linear couplings in the theory. A few examples of sources with different density profiles are explicitly reanalysed in this framework.

Generalized Darmois–Israel Junction Conditions

We present a general method to derive the appropriate Darmois–Israel junction conditions for gravitational theories with higher-order derivative terms by integrating the bulk equations of motion across the singular hypersurface. In higher-derivative theories, the field equations can contain terms which are more singular than the Dirac delta distribution. To handle them appropriately, we formulate a regularization procedure based on representing the delta function as the limit of a sequence of classical functions. This procedure involves imposing suitable constraints on the extrinsic curvature such that the field equations are compatible with the singular source being a delta distribution. As explicit examples of our approach, we demonstrate in detail how to obtain the generalized junction conditions for quadratic gravity, F(R) theories, a 4D low-energy effective action in string theory, and action terms that are Euler densities. Our results are novel, and refine the accuracy of previously claimed results in F(R) theories and quadratic gravity. In particular, when the coupling constants of quadratic gravity are those for the Gauss–Bonnet case, our junction conditions reduce to the known ones for the latter obtained independently by boundary variation of a surface term in the action. Finally, we briefly discuss a couple of applications to thin-shell wormholes and stellar models.

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.