Bilinear Form Research Articles

Multi-wave, breather and interaction solutions to (3+1) dimensional sine-Gordon equation arising in nonlinear physical sciences

Based on the Hirota bilinear forms, the multi-wave solutions of the (3 + 1) dimensional sine-Gordon equation with various physical properties were constructed with using symbolic computation. This equation is characterized by constant coefficients that can be integrated in the Painleve sense. Double exponential and homoclinic breather approaches are used to obtain solutions. In addition, the interaction phenomenon between the exponential function, sine function and the hyperbolic sine function are constructed. For a better understanding the dynamics of the underlying equation three dimensional, density and contour graphs are demonstrated for the appropriately selected parameters. Thus, it has become more possible to observe the physical behaviors of the solutions. The results obtained in this study will contribute to the expansion and enrichment of the solution forms available in the literature of the model under consideration. The results obtained could have important applications in nonlinear optics, solid state physics, optical solitons and quantum field theory.

Abelian groups acting irreducibly and bilinear forms

Abelian groups acting irreducibly and bilinear forms

Calibration estimation of subpopulation total for direct and indirect situations

Sample surveys are conducted to estimate parameters of subpopulation such as mean, variance and totals, etc. The utility of the supporting information lies in the calibration technique, we use to estimate the subpopulation parameter. In this article, we propose calibration-based estimators for a total of the subpopulations. The defined estimators are based on gamma and exponential models. Since both models have right-skewed or positively skewed forms, the defined estimators are appropriate for right-skewed or positively skewed supportive variables. Two situations are considered in which the total of the supporting variable hold for all subpopulations: (i) the total of subpopulations of the supporting variable is known in advance and (ii) the total subpopulations of the supportive variable are not known. Due to difficulty arising in the calibration weight, we have carried out a simulation study in terms of absolute relative bias and simulated relative standard error using Sweden Municipalities 1984 (Sarndal, Swensson, and Wretman, 1992, Appendix B). A simulation study shows that the estimator based on the gamma function is more efficient than ratio, generalized regression, and exponential estimators for all subpopulations in the considered situations.

Influence of a static electric field on magnetic materials monitored by X-ray magnetic circular dichroism

The magnetic circular X-ray dichroism (XMCD) in X-ray absorption spectroscopy (XAS) is extensively used to monitor the response of a magnetic system to an external perturbation. Application of a static external electric field to control its magnetic properties may just lead to a charge rearrangement or to a steady electric current dependent on the geometry. As it is demonstrated, the first situation can be handled with minor modifications of the standard well established schemes to calculate XAS and XMCD spectra. For the second case, however, more advanced schemes have to be used that account for the steady-state out-of-equilibrium situation. It is shown, that this can indeed be achieved by making use of the Keldysh non-equilibrium Green function formalism that allows in particular to express the X-ray absorption coefficient in terms of the corresponding greater Green function G>(E). As an alternative approach, a bi-linear response formalism is presented that can be used to calculate the modification of the XMCD spectra due to the electric field directly.

Line-soliton, lump and interaction solutions to the (2+1)-dimensional Hirota-Satsuma-Ito equation with time-dependent via Hirota bilinear forms

In this paper, we consider the (2+1)-dimensional generalized Hirota-Satsuma-Ito equation with time-dependent, which has applications in describing the propagation of shallow water waves. Based on the bilinear formalism and with the aid of symbolic computation, we obtain line-soliton, lump, one-lump-one-stripe and one-lump-one-soliton using various ansatze’s function. To realize dynamics, we use diverse plots to analyze these solutions. Obtained solutions are reliable in the mathematical physics and engineering.

Study on extensions of (modified) Korteweg–de Vries equations: Painlevé integrability and multiple soliton solutions in fluid mediums

This work develops two higher-dimensional extensions for both Korteweg–de Vries (KdV) and modified KdV (mKdV) equations. We investigate the Painlevé integrability of each couple of the aforementioned two models. We show that the Painlevé integrability fails for one equation of each couple but holds true for the x-derivative of this model. We examine multiple soliton solutions for the integrable extensions of these two models by utilizing the bilinear form. The outcomes will contribute to a deep understanding of the propagation mechanism of the propagation and interaction of multi-solitons in a variety of nonlinear media, including sea waves, optical fibers, and plasma physics.

A novel investigation of extended (3+1)-dimensional shallow water wave equation with constant coefficients utilizing bilinear form

The aim of this research is to investigate new soliton solutions for the shallow water wave equation in an extended (3+1) dimensional space with constant coefficients. The application of this equation is relevant to the analysis of water body dynamics in seas and oceans. Bäcklund transform for the system is constructed from the bilinear form and rational as well exponential traveling wave solutions are analyzed to obtain kink and singular kink solitons. The Hirota bilinear form is further utilized to construct complexiton solutions through extended transformed rational function method. Appropriate parameter values are selected to create 3D and 2D graphical representations, as well as corresponding density plots, for the physical characterization of certain reported results. The obtained results of this research demonstrate how to use discussed approaches for addressing various real world problems described by nonlinear models.

Kernels of Algebraic Curvature Tensors of Symmetric and Skew-Symmetric Builds

The kernel of an algebraic curvature tensor is a fundamental subspace that can be used to distinguish between different algebraic curvature tensors. Kernels of algebraic curvature tensors built only of canonical algebraic curvature tensors of a single build have been studied in detail. We consider the kernel of an algebraic curvature tensor R that is a sum of canonical algebraic curvature tensors of symmetric and skew-symmetric build. An obvious way to ensure that the kernel of R is nontrivial is to choose the involved bilinear forms such that the intersection of their kernels is nontrivial. We present a construction wherein this intersection is trivial but the kernel of R is nontrivial. We also show how many bilinear forms satisfying certain conditions are needed in order for R to have a kernel of any allowable dimension.

Bernhard Riemann 1861 revisited: existence of flat coordinates for an arbitrary bilinear form

We generalize the celebrated results of Bernhard Riemann and Gaston Darboux: we give necessary and sufficient conditions for a bilinear form to be flat. More precisely, we give explicit necessary and sufficient conditions for a tensor field of type (0, 2) which is not necessary symmetric or skew-symmetric, and is possibly degenerate, to have constant entries in a local coordinate system.

Two weight inequality for Hankel form on weighted Bergman spaces induced by doubling weights

The boundedness of the small Hankel operator hfν(g)=Pν(fg‾), induced by an analytic symbol f and the Bergman projection Pν associated to ν, acting from the weighted Bergman space Aωp to Aνq is characterized on the full range 0<p,q<∞ of parameters when ω and ν belong to the class D of radial weights admitting certain two-sided doubling conditions. Moreover, an asymptotic formula for the operator norm of hfν is established in terms of a suitable norm of f(n) depending upon the inducing weights and parameters. Certain results obtained are equivalent to the boundedness of bilinear Hankel forms, which are in turn used to establish the weak factorization Aηq=Aωp1⊙Aνp2, where 1<q,p1,p2<∞ such that q−1=p1−1+p2−1 and η˜1q≍ω˜1p1ν˜1p2. Here τ˜(r)=∫r1τ(t)dt/(1−r) for all 0≤r<1.

Bilinear representation, bilinear Bäcklund transformation, Lax pair and analytical solutions for the fourth-order potential Ito equation describing water waves via Bell polynomials

This paper focuses on the fourth-order nonlinear potential Ito equation, which describes wave patterns in shallow waters. To reveal the integrable characteristics of the considered equation, such as bilinear BT and Lax pair, the Bell polynomials method is used. By employing this technique, the bilinear representation in the form of classical Hirota operators is derived. Moreover, with the help of bilinear form, the bilinear Bäcklund transformation and the Lax pair of the considered equation are obtained successfully. The three wave method in the form of test function is adopted to generate the analytical solutions of the considered equation. By applying this approach, ten analytical solutions are obtained successfully. For each of the obtained solutions, a 3D graph has been plotted by varying free parametric values. These graphs show the different kinds of wave behaviour, including kink-soliton, anti-kink soliton, periodic wave, dark-soliton, bright-soliton, and some complex kink and periodic-type wave solutions.

Complexiton solutions, kink soliton and breather-wave solutions for the (2+1)-dimensional combined potential Kadomtsev-Petviashvili with B-type Kadomtsev-Petviashvili equation

In this paper, the (2+1)-dimensional combined potential Kadomtsev-Petviashvili with B-type Kadomtsev-Petviashvili equation is an object of the research. Based on the extended homogeneous balancing method, Auto-Bäcklund transformations are obtained in two cases. Subsequently, with the help of these transformations, we obtain various explicit solutions of this equation. We attain complexiton solutions consisting of exponential, hyperbolic and trigonometric solutions from the Hirota bilinear form of this equation through the extended transformed rational function method. Also, we derive one-kink and two-kink soliton solutions by Maple symbolic calculation and the breather-wave solution via the extended homoclinic test approach. In addition, 3D graphics and density plots of the obtained solutions are depicted to illustrate the dynamical features of these solutions.

Four-order superconvergent CDG finite elements for the biharmonic equation on triangular meshes

In a conforming discontinuous Galerkin (CDG) finite element method, discontinuous Pk polynomials are employed. To connect discontinuous functions, the inter-element traces, {uh} and {∇uh}, are usually defined as some averages in discontinuous Galerkin finite element methods. But in this CDG finite element method, they are defined as projections of a lifted Pk+4 polynomial from four Pk polynomials on neighboring triangles. With properly chosen weak Hessian spaces, when tested by discontinuous polynomials, the variation form can have no inter-element integral, neither any stabilizer. That is, the bilinear form is the same as that of conforming finite elements for solving the biharmonic equation. Such a conforming discontinuous Galerkin finite element method converges four orders above the optimal order, i.e., the Pk solution has an O(hk+5) convergence in L2-norm, and an O(hk+3) convergence in H2-norm. A local post-process is defined, which lifts the Pk solution to a Pk+4 quasi-optimal solution. Numerical tests are provided, confirming the theory.

Painlevé analysis, multi-kinks and hybrid nonlinear waves for an extended (3 + 1)-dimensional B-type Kadomtsev-Petviashvili equation

In this paper, we study an extended (3+1)-dimensional B-type Kadomtsev-Petviashvili equation having applications in diverse scientific fields. Painlevé anlysis is carried out to test the integrability of the model under consideration. Hirota’s simplified technique is used to investigate one, two and three kink-soliton solutions. Using a dependent variable transformation, bilinear form of the model is obtained which is then used to report lump and lump interaction solutions with periodic and kink waves. The dynamical properties and nature of obtained solutions is comprehensively studied using 3d and 2d graphs.

An online generalized multiscale approximation of the multipoint flux mixed finite element method

In this paper, we develop an online generalized multiscale approximation of the multipoint flux mixed finite element method (MFMFE) for Darcy flow in highly heterogeneous porous media, which can handle full tensor permeability fields and unstructured grids with local velocity elimination. Here, for illustration of the proposed method, we will mainly consider the situation that the porous medium is filled with fractures, which usually results in unstructured grids, and the extension to the situation that the permeability is represented as a full tensor is straightforward. From the fine-grid discretization by the MFMFE method, we derive the symmetric and positive definite discrete bilinear form for both the matrix and fracture pressure, and thereby obtain the corresponding discrete weak formulation that only related to pressure. In the offline stage, we compute offline basis functions which contain the important local multiscale information of each coarse-grid block to form the initial multiscale space. In the online stage, we construct and add online basis functions to enrich the multiscale space and consequently improve the accuracy of the multiscale solution. Each offline basis function and each online basis function contains multiscale information both for matrix and fracture. We give the theoretical analysis for the convergence of the online enrichment, which shows that more sufficient initial basis functions lead to faster convergence rates. A series of numerical examples that either the permeability is a full tensor or the porous medium is filled with fractures are presented to show the performance of the multiscale method.

Study on the (2+1)-dimensional extension of Hietarinta equation: soliton solutions and Bäcklund transformation

The (1+1)-dimensional bilinear Hietarinta equation was firstly proposed when searching for integrable nonlinear evolution equations by the three-soliton method. In this paper, we focus on the (2+1)-dimensional extension of Hietarinta equation, which enjoys potential application in environmental engineering. Based on the bilinear form, one-soliotn and two-soliton solutions are derived. Bilinear Bäcklund transformation and Bell-polynomial-typed Bäcklund transformation are derived through the Hirota bilinear method and Bell polynomials, respectively. The three-dimensional plots of soliton solutions have been given by selecting appropriate parameters.

Solitary waves for the nonparaxial nonlinear Schrödinger equation

In this paper, the nonparaxial nonlinear Schrödinger (NNLS) equation by considering its integrability which enables the propagation of ultra-broad nonparaxial beams in a planar optical waveguide is studied. The plenty numbers of solitary wave solutions by using Hirota’s bilinear scheme are found, in addition, the bilinear transformation and also the related theorem for getting to the bilinear form of nonlinear system are considered. Two new simple approaches are implemented to recover periodic wave, bright soliton, singular, and singular soliton for this model. Because of the significance of the NNLS in modeling the propagation of solitons through an optical fiber, the recovered solitons are vital for describing and understanding a variety of fundamental physical processes. The effect of the free parameters on the behavior of acquired figures to a few obtained solutions by providing the feasibility and reliability of the used procedure was also discussed. For more physical illustration and knowledge of the physical characteristics of this equation, some important solutions are discussed graphically in the form of 2D and 3D plots by selecting suitable parameters.

Breather, soliton molecules, soliton fusions and fissions, and lump wave of the Caudrey-Dodd-Gibbon equation

In this article, our attention is focused on the exploration of new features of the Caudrey-Dodd-Gibbon (CDG) equation arising from fluid mechanism. We introduce a constant in the transformation, which links the solution and auxiliary function defined in the bilinear form. By constructing different auxiliary function, we calculate the breather solution, one- to three-soliton solutions and lump wave solution. We report that a breather can be generated from a stripe-like soliton. We discover the soliton molecules and their interaction where the maximum amplitude will decrease as they overlap. Two types of heterotypic solitons, namely, soliton fusions and fissions are obtained by attaining their constrain conditions, respectively. We also observe this equation possesses several unique features, such as, having only the two-soliton molecules but not N (N ≥ 3)-soliton molecules, and having the line-like lump wave parallel to the x-axis but not to the t-axis.

Dynamics of breather, multi-wave, interaction and other wave solutions to the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

PurposeThe purpose of this paper is to study the new (3 + 1)-dimensional integrable fourth-order nonlinear equation which is used to model the shallow water waves.Design/methodology/approachBy means of the Cole–Hopf transform, the bilinear form of the studied equation is extracted. Then the ansatz function method combined with the symbolic computation is implemented to construct the breather, multiwave and the interaction wave solutions. In addition, the subequation method tis also used to search for the diverse travelling wave solutions.FindingsThe breather, multiwave and the interaction wave solutions and other wave solutions like the singular periodic wave structure and dark wave structure are obtained. To the author’s knowledge, the solutions obtained are all new and have never been reported before.Originality/valueThe solutions obtained in this work have never appeared in other literature and can be regarded as an extension of the solutions for the new (3 + 1)-dimensional integrable fourth-order nonlinear equation.

Quadratic Lie algebras with 2-plectic structures

We study the existence of 2-plectic structures on Lie algebras which admit an ad-invariant non-degenerate symmetric bilinear form, frequently called quadratic Lie algebras. It is well-known that every centerless quadratic Lie algebra admits a 2-plectic form but not many quadratic examples with nontrivial center are known. In this paper we give several constructions to obtain large families of 2-plectic quadratic Lie algebras with nontrivial center, many of them among the class of nilpotent Lie algebras. We give some sufficient conditions to assure that certain extensions of 2-plectic quadratic Lie algebras result to be 2-plectic as well. We prove that every quadratic and symplectic Lie algebra with dimension greater than 4 also admits a 2-plectic form. Further, conditions to assure that one may find a 2-plectic which is exact on certain quadratic Lie algebras are also obtained.