Zakharov–Shabat system and hyperbolic pseudoanalytic function theory

Kravchenko, Viktor G.; Kravchenko, Vladislav V.; Tremblay, Sébastien

doi:10.1002/mma.1206

Cited by 5 publications

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“…where a and b are bicomplex functions of the variable z ∈ D. In [20], [23], [25] it was shown that many results from pseudoanalytic function theory [2] remain valid in the hyperbolic situation. We will not give here definitions and properties corresponding to the general Vekua equation (2.3) referring the reader to [23].…”

Section: Hyperbolic Vekua Equationmentioning

confidence: 99%

“…We use our recent result on the construction of the kernel of the transmutation operator corresponding to a Darboux associated Schrödinger operator [26] to find out that a bicomplex-valued function whose one complex component is the transmutation kernel K 1 (x, t) (for a Schrödinger operator d 2 dx 2 − q 1 (x), q 1 ∈ C[−b, b]) and the other complex component is K 2 (x, t) (for a Schrödinger operator d 2 dx 2 − q 2 (x), with q 2 obtained from q 1 by a Darboux transformation) is a solution of a certain hyperbolic Vekua equation of a special form (for the theory of elliptic Vekua equations see [47] as well as [2] and [23]). In spite of a recent progress reported in [20], [23], [25], [30] the theory of hyperbolic Vekua equations is considerably less developed than the theory of classical (elliptic) Vekua equations. For example, as was shown in [25] (see also [23]) the construction of so-called formal powers (solutions of the Vekua equation generalizing the usual complex powers (z − z 0 ) n ) can be performed using the definitions completely analogous to those introduced by L. Bers [2].…”

Section: Introductionmentioning

confidence: 99%

“…In spite of a recent progress reported in [20], [23], [25], [30] the theory of hyperbolic Vekua equations is considerably less developed than the theory of classical (elliptic) Vekua equations. For example, as was shown in [25] (see also [23]) the construction of so-called formal powers (solutions of the Vekua equation generalizing the usual complex powers (z − z 0 ) n ) can be performed using the definitions completely analogous to those introduced by L. Bers [2].…”

Section: Introductionmentioning

confidence: 99%

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Construction of Transmutation Operators and Hyperbolic Pseudoanalytic Functions

Kravchenko

Torba

2014

Complex Anal. Oper. Theory

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A representation for integral kernels of Delsarte transmutation operators is obtained in the form of a functional series with the exact formulae for the terms of the series. It is based on the application of hyperbolic pseudoanalytic function theory and recent results on mapping properties of the transmutation operators.The kernel K1 of the transmutation operator relating A = − d 2 dx 2 + q1(x) and B = − d 2 dx 2 results to be one of the complex components of a bicomplex-valued hyperbolic pseudoanalytic function satisfying a Vekua-type hyperbolic equation of a special form. The other component of the pseudoanalytic function is the kernel of the transmutation operator relating C = − d 2 dx 2 + q2(x) and B where q2 is obtained from q1 by a Darboux transformation. We prove the expansion theorem and a Runge-type theorem for this special hyperbolic Vekua equation and using several known results from hyperbolic pseudoanalytic function theory together with the recently discovered mapping properties of the transmutation operators obtain the new representation for their kernels. Several examples are given. Moreover, based on the presented results approaches for numerical computation of the transmutation kernels and for numerical solution of spectral problems are proposed.

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Section: Hyperbolic Vekua Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Construction of Transmutation Operators and Hyperbolic Pseudoanalytic Functions

Kravchenko

Torba

2014

Complex Anal. Oper. Theory

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“…In 2008, Kravchenko et. al. show in [27] and [28] how hyperbolic numbers to analyze certain solutions to the Klein-Gordon equation. In 2014, Terlizzi, Konderak and Lacirasella provide many explicit formulae for functions over Lorentz numbers which they use to study manifolds modelled over Lorentz numbers [44].…”

Section: Historymentioning

confidence: 99%

Introduction to $\mathcal{A}$-Calculus

Cook¹

2017

Preprint

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Let A denote an n-dimensional associative algebra over R. This paper gives an introductory exposition of calculus over A. An A-differentiable function f : A → A is one for which the differential is right-A-linear. The basis-dependent correspondence between right-A-linear maps and the regular representation of real matrices is discussed in detail. The requirement that the Jacobian matrix of a function fall in the regular representation of A gives n 2 −n generalized A-CR equations. In contrast, some authors use a deleted-difference quotient to describe differentiability over an algebra. We compare these concepts of differentiability over an algebra and prove they are equivalent in the semisimple commutative case. We also show how difference quotients are ill-equipt to study calculus over a nilpotent algebra.Cauchy Riemann equations are elegantly captured by the Wirtinger calculus as ∂f ∂ z = 0. We generalize this to any commutative unital associative algebra. Instead of one conjugate, we need n − 1 conjugate variables. Our construction modifies that given by Alvarez-Parrilla, Frías-Armenta, López-González and Yee-Romero in [16].We derive Taylor's Theorem over an algebra. Following Wagner, we show how Generalized Laplace equations are naturally seen from the multiplication table of an algebra. We show how d'Alembert's solution to the wave-equation can be derived from the function theory of an appropriate algebra. The inverse problem of A-calculus is introduced and we show how the Tableau for an A-differentiable function has a rather special form. The integral over an algebra is also studied. We find the usual elementary topological results about closed and exact forms generalize nicely to our integral. Generalizations of the First and Second Fundamental Theorems of Calculus are given. This paper should be accessible to undergraduates with a firm foundation in linear algebra and calculus. Some background in complex analysis would also be helpful.

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“…In [10] and [4] "hyperbolic pseudoanalytic function theory" was studied where hyperbolic numbers D (also called duplex numbers) [16] defined by D := x + tj : j 2 = 1, x, t ∈ R ∼ = Cl R (0, 1) are considered instead of (elliptic) complex numbers. Here we show that the concept of the transplant operator can be introduced in this context as well and it allows one to solve hyperbolic main Vekua equations related to the Klein-Gordon equation.…”

Section: Hyperbolic Pseudoanalytic Function Theorymentioning

confidence: 99%

Explicit solutions of generalized Cauchy–Riemann systems using the transplant operator

Kravchenko

Tremblay

2010

Journal of Mathematical Analysis and Applications

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In Kravchenko (2008) [8] it was shown that the tool introduced there and called the transplant operator transforms solutions of one Vekua equation into solutions of another Vekua equation, related to the first via a Schrödinger equation. In this paper we prove a fundamental property of this operator: it preserves the order of zeros and poles of generalized analytic functions and transforms formal powers of the first Vekua equation into formal powers of the same order for the second Vekua equation. This property allows us to obtain positive formal powers and a generating sequence of a "complicated" Vekua equation from positive formal powers and a generating sequence of a "simpler" Vekua equation. Similar results are obtained regarding the construction of Cauchy kernels. Elliptic and hyperbolic pseudoanalytic function theories are considered and examples are given to illustrate the procedure.

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Zakharov–Shabat system and hyperbolic pseudoanalytic function theory

Cited by 5 publications

References 9 publications

Construction of Transmutation Operators and Hyperbolic Pseudoanalytic Functions

Construction of Transmutation Operators and Hyperbolic Pseudoanalytic Functions

Introduction to $\mathcal{A}$-Calculus

Explicit solutions of generalized Cauchy–Riemann systems using the transplant operator

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