Quaternionic Equation for Electromagnetic Fields in Inhomogeneous Media

Kravchenko, Vladislav V.

doi:10.1142/9789812794253_0042

Cited by 37 publications

(84 citation statements)

References 10 publications

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“…The generalization of this fact for a multidimensional situation was obtained in [8] (see also [10]). Among the peculiar properties of the Riccati equation stands out an important theorem of Euler, dating from 1760.…”

Section: Introductionmentioning

confidence: 88%

“…Thus, given a particular solution of the quaternionic Riccati equation, the general solution reduces to the linear first order equation (10), exactly as in one-dimensional situation.…”

Section: Lemma 4 [9]mentioning

confidence: 99%

“…Given a nonvanishing particular solution f 0 of the Schrödinger equation (4) we construct the vector h = Df 0 /f 0 and consider equation (10). Taking a solution F of (10) we find Ψ from (18).…”

Section: Remarkmentioning

confidence: 99%

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

Kravchenko¹

2005

J. Phys. A: Math. Gen.

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Given a particular solution of a one-dimensional stationary Schrödinger equation this equation of second order can be reduced to a first order linear ordinary differential equation. This is done with the aid of an auxiliary Riccati differential equation. In the present work we show that the same fact is true in a multidimensional situation also. For simplicity we consider the case of two or three independent variables. One particular solution of the Schrödinger equation 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 quaternionic Riccati equation. The resulting first order quaternionic equation is equivalent to the static Maxwell system and is closely related to the Dirac equation. In the case of two independent variables it is the well known Vekua equation from theory of pseudoanalytic (or generalized analytic) functions. Nevertheless we show that even in this case it is very useful to consider not complex valued functions only, solutions of the Vekua equation but complete quaternionic functions. In this way the first order quaternionic equation represents two separate Vekua equations, one of which gives us solutions of the Other results from theory of pseudoanalytic functions can be written down for solutions of the Schrödinger equation. Moreover, for an ample class of potentials in the Schrödinger equation (which includes for instance all radial potentials), this new approach gives us a simple procedure allowing to obtain an infinite sequence of solutions of the Schrödinger equation from one known particular solution.

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Section: Introductionmentioning

confidence: 88%

“…Thus, given a particular solution of the quaternionic Riccati equation, the general solution reduces to the linear first order equation (10), exactly as in one-dimensional situation.…”

Section: Lemma 4 [9]mentioning

confidence: 99%

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

Kravchenko¹

2005

J. Phys. A: Math. Gen.

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“…More information on the structure of the algebra of complex quaternions can be found for example in [16] or [21].…”

Section: Explicit Construction Of Positive Formal Powersmentioning

confidence: 99%

On a factorization of second-order elliptic operators and applications

Kravchenko¹

2006

J. Phys. A: Math. Gen.

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We show that given a nonvanishing particular solution of the equation (div p grad +q)u = 0,the corresponding differential operator can be factorized into a product of two first order operators. The factorization allows us to reduce the equation (1) to a first order equation which in a two-dimensional case is the Vekua equation of a special form. Under quite general conditions on the coefficients p and q we obtain an algorithm which allows us to construct in explicit form the positive formal powers (solutions of the Vekua equation generalizing the usual powers (z − z0) n , n = 0, 1, . . .). This result means that under quite general conditions one can construct an infinite system of exact solutions of (1) explicitly, and moreover, at least when p and q are real valued this system will be complete in ker(div p grad +q) in the sense that any solution of (1) in a simply connected domain Ω can be represented as an infinite series of obtained exact solutions which converges uniformly on any compact subset of Ω. Finally we give a similar factorization of the operator (div p grad +q) in a multidimensional case and obtain a natural generalization of the Vekua equation which is related to second order operators in a similar way as its two-dimensional prototype does.

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“…Following e.g. References [3][4][5], this then opened the door to treat further related PDEs such as for example time-dependent Maxwell's equations or the Schr odinger equation.…”

Section: Introductionmentioning

confidence: 99%

Further results on the asymptotic growth of entire solutions of iterated Dirac equations in ℝn

Constales

Almeida

Kraußhar

2006

Math. Meth. Appl. Sci.

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SUMMARYIn this paper, we establish some further results on the asymptotic growth behaviour of entire solutions to iterated Dirac equations in R n . Solutions to this type of systems of partial di erential equations are often called polymonogenic or k-monogenic. In the particular cases where k is even, one deals with polyharmonic functions. These are of central importance for a number of concrete problems arising in engineering and physics, such as for example in the case of the biharmonic equation for the description of the stream function in the Stokes ow regime with low Reynolds numbers and for elasticity problems in plates.The asymptotic study that we are going to perform within the context of these PDE departs from the Taylor series representation of their solutions. Generalizations of the maximum term and the central index serve as basic tools in our analysis. By applying these tools we then establish explicit asymptotic relations between the growth behaviour of polymonogenic functions, the growth behaviour of their iterated radial derivatives and that of functions obtained by applying iterations of the operator to them.

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Quaternionic Equation for Electromagnetic Fields in Inhomogeneous Media

Abstract: We show that the Maxwell equations for arbitrary inhomogeneous media are equivalent to a single quaternionic equation which can be considered as a generalization of the Vekua equation for generalized analytic functions.

Cited by 37 publications

References 10 publications

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

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

On a factorization of second-order elliptic operators and applications

Further results on the asymptotic growth of entire solutions of iterated Dirac equations in ℝn

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