Quaternionic Equation Research Articles

Quaternionic-valued ordinary differential equations. The Riccati equation

We give some sufficient conditions for the existence of at least two periodic solutions of the quaternionic Riccati equation. In some cases we are able to give a full description of dynamics and detect solutions heteroclinic to the periodic ones. We also provide an example of the Riccati equation without periodic solutions which appears in the Euler vorticity dynamics.

Classification of diffeomorphisms of $\mathbb S^4$ induced by queternionic Riccati equations with periodic coefficients

The monodromy maps for the quaternionic Riccati equations with periodic coefficients $\dot{z}=zp(t)z+q(t)z+zr(t)+s(t)$ in $\mathbb H\mathbb P^{1}$ are quternionic Möbius transformations. We prove that, like in the case of automorphisms of $\mathbb C\mathbb P^{1}$, the quaternionic homografies are divided into three classes: hyperbolic, elliptic and parabolic.

Cayley Formula, Euler Parameters and Rotations in 3-Dimensional Lorentzian Space

In this study, by using Lorentzian matrix multiplication, Cayley formula and Euler parameters of a Lorentzian orthogonal matrix are obtained in Lorentz space L3. Then, by using Euler parameters of a rotation in a split quaternion, the split quaternion equation of a rotation movement is obtained in the space L3.

Skew-Symmetric Algorithm for Inverting NxN Quaternion Equations Using NxN Real Variable Matrix Arithmetic

Quaternion matrix inverses and linear equation solutions are often computed by transforming a given A ∈ QN xN quaternion matrix into an equivalent \(A \in {\cal R}^{4N x4N}\) real matrix. The transformation process is well-known, but as N becomes large the computational costs increase rapidly. This paper presents a multi-level skew-symmetric partitioning algorithm for the \({\cal R}^{4N x4N}\) matrix representation that requires only four \({\cal R}^{4N x4N}\) matrix inversions to complete the solution. A two-stage algorithm is presented. First the \({\cal R}^{4N x4N}\) matrix is partitioned into four blocks using two \({\cal R}^{2N x24N}\) matrices. A partitioned matrix inverse is presented for inverting each \({\cal R}^{2N x2N}\) matrix partition. A second level of the partitioning exploits the skew-symmetric sub-structure of the \({\cal R}^{2N x2N}\) partitioned matrix inverse solutions. The proposed algorithm improves the computational performance ∼ 2X and minimizes the memory requirements by only requiring processing for four \({\cal R}^{N xN}\) matrix partitions, when compared to standard algorithms for inverting the \({\cal R}^{4N x4N}\) matrix. Example applications are presented for a purely quaternion matrix inversion algorithm, as well as solution algorithms for the \({\cal R}^{2N x2N}\) and \({\cal R}^{4N x4N}\) real variable versions of the algorithms.

Solving Two Pairs of Quadratic Equations in Quaternions

Two quaternions a and b are said to be semi-similar if there are x and y such that xay = b and yax = b; said to be con-semi-similar if there are x and y such that \(\bar{x}ay = b\) and \(\bar{y}bx = a\). In this note, the solvability conditions and general solutions to the two pairs of equations in quaternions are established.

Relativistic quaternionic wave equation. II

Further results are reported for the one-component quaternionic wave equation recently introduced. A Lagrangian is found for the momentum-space version of the free equation, and another, nonlocal in time, is found for the complete equation. Further study of multiparticle systems has us looking into the mathematics of tensor products of Hilbert spaces. The principles of linearity and superposition are also clarified to good effect in advancing the quaternionic theory.

Relativistic quaternionic wave equation

We study a one-component quaternionic wave equation which is relativistically covariant. Bilinear forms include a conserved four-vector current and an antisymmetric second rank tensor. Waves propagate within the light cone and there is a conserved quantity which looks like helicity. The principle of superposition is retained in a slightly altered manner. External potentials can be introduced in a way that allows for gauge invariance. There are some results for scattering theory and for two-particle wave functions as well as the beginnings of second quantization. However, we are unable to find a suitable Lagrangian or an energy-momentum tensor.

Solving Linear and Quadratic Quaternion Equations

Solving Linear and Quadratic Quaternion Equations

Observability analysis of rotation estimation by fusing inertial and line-based visual information: A revisit

This note presents a simple approach to the observability analysis of the rotation estimation using line-based dynamic vision and inertial sensors. The problem was originally raised and formulated in Rehbinder, and Ghosh [2003. Pose estimation using line-based dynamic vision and inertial sensors. IEEE Transactions on Automatic Control, 48(2), 186–199.] where the unobservable subgroup was derived using complex matrix manipulations. By solving linear quaternion equations and using set operations, we not only successfully obtain the same result but also naturally extend it for the case with linearly dependent lines. The development in this note is more straightforward and gives rise to a clearer picture of the problem.

Cramer rule for quaternionic linear equations in quaternionic quantum theory

By means of a complex representation of a quaternion matrix and a companion vector, this paper introduces a new definition of determinant for a quaternion matrix, derives a technique of finding an inverse matrix of a quaternion invertible matrix, and gives a Cramer rule for quaternionic linear equations in quaternionic quantum theory.

Zeros of unilateral quaternionic polynomials

The purpose of this paper is to show how the problem of finding the zeros ofunilateral n-order quaternionic polynomials can be solved by determining the eigenvectors of the corresponding companion matrix. This approach, probably superfluous in the case of quadratic equations for which a closed formula can be given, becomes truly useful for (unilateral) n-order polynomials. To understand the strength of this method, it is compared with the Niven algorithm and it is shown where this (full) matrix approach improves previous methods based on the use of the Niven algorithm. For convenience of the readers, some examples of second and third order unilateral quaternionic polynomials are explicitly solved. The leading idea of the practical solution method proposed in this work can be summarized in the following three steps: translating the quaternionic polynomial in the eigenvalue problem for its companion matrix, finding its eigenvectors, and, finally, giving the quaternionic solution of the unilateral polynomial in terms of the components of such eigenvectors. A brief discussion on bilateral quaternionic quadratic equations is also presented.

Algebraic tools for the study of quaternionic behavioral systems

In this paper we study behavioral systems whose trajectories are given as solutions of quaternionic difference equations. As happens in the commutative case, it turns out that quaternionic polynomial matrices play an important role in this context. Therefore we pay special attention to such matrices and derive new results concerning their Smith form. Based on these results, we obtain a characterization of system theoretic properties such as controllability and stability of a quaternionic behavior.

On the reduction of the multidimensional stationary Schrödinger equation to a first-order equation and its relation to the pseudoanalytic function theory

Given a particular solution of a one-dimensional stationary Schroedinger equation (SE) this equation of second order can be reduced to a first order linear differential equation. This is done with the aid of an auxiliary Riccati equation. We show that a similar fact is true in a multidimensional situation also. We consider the case of two or three independent variables. One particular solution of (SE) allows us to reduce this second order equation to a linear first order quaternionic differential equation. As in one-dimensional case this is done with the aid of an auxiliary Riccati equation. The resulting first order quaternionic equation is equivalent to the static Maxwell system. In the case of two independent variables it is the Vekua equation from theory of generalized analytic functions. We show that even in this case it is necessary to consider not complex valued functions only, solutions of the Vekua equation but complete quaternionic functions. Then the first order quaternionic equation represents two separate Vekua equations, one of which gives us solutions of (SE) and the other can be considered as an auxiliary equation of a simpler structure. For the auxiliary equation we always have the corresponding Bers generating pair, the base of the Bers theory of pseudoanalytic functions, and what is very important, the Bers derivatives of solutions of the auxiliary equation give us solutions of the main Vekua equation and as a consequence of (SE). We obtain an analogue of the Cauchy integral theorem for solutions of (SE). For an ample class of potentials (which includes for instance all radial potentials), this new approach gives us a simple procedure allowing to obtain an infinite sequence of solutions of (SE) from one known particular solution.

Real linear quaternionic differential operators

The renewed interest in searching for quaternionic deviations of standard (complex) quantum mechanics resulted, in the last years, in a better understanding of the quaternionic mathematical tools needed to solve quantum mechanical problems. In particular, a relevant progress has been achieved in solving eigenvalue problems and differential equations for quaternionic operators. The practical methods recently proposed to solve quaternionic and complex linear second-order differential equations with constant coefficients represent a fundamental starting point to discuss quaternionic potentials in quantum mechanics and study possible violations from complex theories. Nevertheless, only for a restricted class of real linear quaternionic differential operators (namely, symmetric operators) the solution of differential problems was given. In this paper, we study real linear quaternionic differential equations. The proposed resolution's method is based on the Jordan canonical form of (real linear) quaternionic matrices.

An algorithm for quaternionic linear equations in quaternionic quantum theory

By means of complex representation and companion vector, in this paper we introduce a definition of rank of a quaternion matrix, study the problems of quaternionic linear equations, and obtain an algorithm for quaternionic linear equations in quaternionic quantum theory.

On a quaternionic Maxwell equation for the time-dependent electromagnetic field in a chiral medium

Maxwell's equations for the time-dependent electromagnetic field in a homogeneous chiral medium are reduced to a single quaternionic equation. Its fundamental solution satisfying the causality principle is obtained which allows us to solve the time-dependent chiral Maxwell system with sources.

Quaternionic factorization of the Schrödinger operator and its applications to some first-order systems of mathematical physics

We consider the following first order systems of mathematical physics. 1.The Dirac equation with scalar potential. 2.The Dirac equation with electric potential. 3.The Dirac equation with pseudoscalar potential. 4.The system describing non-linear force free magnetic fields or Beltrami fields with nonconstant proportionality factor. 5.The Maxwell equations for slowly changing media. 6.The static Maxwell system. We show that all this variety of first order systems reduces to a single quaternionic equation the analysis of which in its turn reduces to the solution of a Schroedinger equation with biquaternionic potential. In some important situations the biquaternionic potential can be diagonalized and converted into scalar potentials.

Quaternionic Madelung transformation and non-Abelian fluid dynamics

In the 1920's, Madelung noticed that if the complex Schroedinger wavefunction is expressed in polar form, then its modulus squared and the gradient of its phase may be interpreted as the hydrodynamic density and velocity, respectively, of a compressible fluid. In this paper, we generalize Madelung's transformation to the quaternionic Schroedinger equation. The non-abelian nature of the full SU(2) gauge group of this equation leads to a richer, more intricate set of fluid equations than those arising from complex quantum mechanics. We begin by describing the quaternionic version of Madelung's transformation, and identifying its ``hydrodynamic'' variables. In order to find Hamiltonian equations of motion for these, we first develop the canonical Poisson bracket and Hamiltonian for the quaternionic Schroedinger equation, and then apply Madelung's transformation to derive non-canonical Poisson brackets yielding the desired equations of motion. These are a particularly natural set of equations for a non-abelian fluid, and differ from those obtained by Bistrovic et al. only by a global gauge transformation. Because we have obtained these equations by a transformation of the quaternionic Schroedinger equation, and because many techniques for simulating complex quantum mechanics generalize straightforwardly to the quaternionic case, our observation leads to simple algorithms for the computer simulation of non-abelian fluids.

Solving simple quaternionic differential equations

The renewed interest in investigating quaternionic quantum mechanics, in particular tunneling effects, and the recent results on quaternionic differential operators motivate the study of resolution methods for quaternionic differential equations. In this paper, by using the real matrix representation of left/right acting quaternionic operators, we prove existence and uniqueness for quaternionic initial value problems, discuss the reduction of order for quaternionic homogeneous differential equations and extend to the noncommutative case the method of variation of parameters. We also show that the standard Wronskian cannot uniquely be extended to the quaternionic case. Nevertheless, the absolute value of the complex Wronskian admits a noncommutative extension for quaternionic functions of one real variable. Linear dependence and independence of solutions of homogeneous (right) H-linear differential equations is then related to this new functional. Our discussion is, for simplicity, presented for quaternionic second order differential equations. This involves no loss of generality. Definitions and results can be readily extended to the n-order case.

An operator method for finding exact solutions to vector Korteweg–de Vries equations

We develop an operator method which helps finding exact solutions to nonlinear evolution equations (NLEs). Our working schema goes as follows: First we translate the given (NLE) into an appropriate operator version (ONLE). Second, we look for solutions to (ONLE) of the form U=(I+L)−1M, where both L and M are operator-valued functions of the space–time variables and the range of M locates in some appropriate Banach algebras which admits a functional φ that preserves the squares [i.e., φ(A2)=φ(A)2]. Finally, a solution u of the given (NLE) can be obtained by setting u≔φ(U). This method is named by the LM method. Using the LM method, we have rederived the famous Cole–Hopf transformation which reduces the nonlinear Burgers equation into the linear heat equation. The main part of this article is to use the LM method to study the vector Korteweg–de Vries (KdV) equations ut=uxxx+3(u2)x settled in finite-dimensional unital Banach algebras 𝒥. It is shown that these vector KdV equations admit soliton solutions. Specially, we have carried out a thorough study of the quaternionic KdV equation (i.e., the vector KdV equation settled in the Hamilton quaternion algebra ℍ) and shown many interesting and surprising aspects of the quaternionic KdV solitons. Two of them read as follows. (a) The paradoxical energy symmetry breaking phenomenon: Two quaternionic KdV solitons with different energies can annihilate each other. (b) The surprising low-dimensional phenomenon: The interaction of any finitely many quaternionic KdV solitons which live in a unital three-dimensional subspace Π of ℍ does not yield any effect to the part outside that subspace Π and thus their interaction behaves as if it were linear although the interaction between quaternionic KdV solitons is really nonlinear. The LM method can be thought as a complement to the famous bilinear operator method of Hirota. Hirota’s method works very powerful for solving scalar equation but has difficulty with vector equations. The LM method helps overcoming this difficulty.