Fundamentals of Bicomplex Pseudoanalytic Function Theory: Cauchy Integral Formulas, Negative Formal Powers and Schrödinger Equations with Complex Coefficients

The study of the Dirac system and second-order elliptic equations with complex-valued coefficients on the plane naturally leads to bicomplex Vekua-type equations (Campos et al. in Adv Appl Clifford Algebras, 2012; Castañeda et al. in J Phys A Math Gen 38:9207–9219, 2005; Kravchenko in J Phys A Math Gen 39:12407–12425, 2006). To the difference of complex pseudoanalytic (or generalized analytic) functions (Bers in Theory of pseudo-analytic functions. New York University, New York, 1952; Vekua in Generalized analytic functions. Nauka, Moscow (in Russian); English translation Oxford, 1962. Pergamon Press, Oxford, 1959) the theory of bicomplex pseudoanalytic functions has not been developed. Such basic facts as, e.g., the similarity principle or the Liouville theorem in general are no longer available due to the presence of zero divisors in the algebra of bicomplex numbers. In the present work we develop a theory of bicomplex pseudoanalytic formal powers analogous to the developed by Bers (Theory of pseudo-analytic functions. New York University, 1952) and especially that of negative formal powers. Combining the approaches of Bers and Vekua with some additional ideas we obtain the Cauchy integral formula in the bicomplex setting. In the classical complex situation this formula was obtained under the assumption that the involved Cauchy kernel is global, a very restrictive condition taking into account possible practical applications, especially when the equation itself is not defined on the whole plane. We show that the Cauchy integral formula remains valid with the Cauchy kernel from a wider class called here the reproducing Cauchy kernels. We give a complete characterization of this class. To our best knowledge these results are new even for complex Vekua equations. We establish that reproducing Cauchy kernels can be used to obtain a full set of negative formal powers for the corresponding bicomplex Vekua equation and present an algorithm which allows one their construction. Bicomplex Vekua equations of a special form called main Vekua equations are closely related to stationary Schrödinger equations with complex-valued potentials. We use this relation to establish useful connections between the reproducing Cauchy kernels and the fundamental solutions for the Schrödinger operators which allow one to construct the Cauchy kernel when the fundamental solution is known and vice versa. Moreover, using these results we construct the fundamental solutions for the Darboux transformed Schrödinger operators.     

On the Vekua Pair in Spheroidal Geometry and its Role in Solving Boundary Value Problems

The Vekua pair forms a transformation between the kernel of the Laplace's and the kernel of the Helmholtz's operator. In fact, it provides an interior solution of the Helmholtz's equation once an interior harmonic function is given, and conversely, given an interior solution of the Helmhotz's equation an interior harmonic function is constructed. Consequently, it seems that the Vekua connection offers the perfect ground to obtain solutions of boundary value problems connected with Helmholtz operator. Vekua expressed his transformation in spherical coordinates. Nevertheless, when a change of coordinates is applied, the transformation assumes a much more complicated form, but it still remains a very useful technique for dealing with solutions of the equations of Laplace and Helmholtz. Here we extend the Vekua theory to a new integral transformation pair concerning solutions of the aforementioned operators in exterior domains. In addition, the form of the Vekua transformation is analyzed in spheroidal coordinates and its implication to boundary value problems is investigated.     

Conservation laws and the variational bicomplex for second-order scalar hyperbolic equations in the plane

In this paper, we announce several new results concerning the cohomology of the variational bicomplex for a second-order scalar hyperbolic equation in the plane. These cohomology groups are represented by the conservation laws, and certain form-valued generalizations, for the equation. Our methods are based upon the introduction of an adapted coframe for the the variational bicomplex which is constructed by generalizing the classical Laplace transformation used to integrate certain linear hyperbolic equations in the plane.     

Future Stability of the Einstein-Maxwell-Scalar Field System

Ringstr?m managed (in Invent Math 173(1):123–208, 2008) to prove future stability of solutions to Einstein’s field equations when matter consists of a scalar field with a potential creating an accelerated expansion. This was done for a quite wide class of spatially homogeneous space–times. The methods he used should be applicable also when other kinds of matter fields are added to the stress-energy tensor. This article addresses the question whether we can obtain stability results similar to those Ringstr?m obtained if we add an electromagnetic field to the matter content. Before this question can be addressed, more general properties concerning Einstein’s field equation coupled to a scalar field and an electromagnetic field have to be settled. The most important of these questions are the existence of a maximal globally hyperbolic development and the Cauchy stability of solutions to the initial value problem.     

Vekua theory for the Helmholtz operator

Vekua operators map harmonic functions defined on domain in \({\mathbb R^{2}}\) to solutions of elliptic partial differential equations on the same domain and vice versa. In this paper, following the original work of I. Vekua (Ilja Vekua (1907–1977), Soviet-Georgian mathematician), we define Vekua operators in the case of the Helmholtz equation in a completely explicit fashion, in any space dimension N ≥ 2. We prove (i) that they actually transform harmonic functions and Helmholtz solutions into each other; (ii) that they are inverse to each other; and (iii) that they are continuous in any Sobolev norm in star-shaped Lipschitz domains. Finally, we define and compute the generalized harmonic polynomials as the Vekua transforms of harmonic polynomials. These results are instrumental in proving approximation estimates for solutions of the Helmholtz equation in spaces of circular, spherical, and plane waves.     

Vekua theory for the Helmholtz operator

Vekua operators map harmonic functions defined on domain in \mathbb R²{\mathbb R^{2}} to solutions of elliptic partial differential equations on the same domain and vice versa. In this paper, following the original work of I. Vekua (Ilja Vekua (1907–1977), Soviet-Georgian mathematician), we define Vekua operators in the case of the Helmholtz equation in a completely explicit fashion, in any space dimension N ≥ 2. We prove (i) that they actually transform harmonic functions and Helmholtz solutions into each other; (ii) that they are inverse to each other; and (iii) that they are continuous in any Sobolev norm in star-shaped Lipschitz domains. Finally, we define and compute the generalized harmonic polynomials as the Vekua transforms of harmonic polynomials. These results are instrumental in proving approximation estimates for solutions of the Helmholtz equation in spaces of circular, spherical, and plane waves.     

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

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.     

Measure diffusions and related explosion problems

For random measure-valued stochastic partial differential equations for biological processes, growth represented by scalar partial differential equations at each point of the support and spread being a diffusion on R ^d, solutions are constructed by smearing the growth processes at each spatial point and composing the resulting generator with the generator for the spread. If these solutions are unique the equation is called solvable. We find conditions for the noise term of a solvable equations to have trivial effect and we identify some non-solvable equations, for example the diffusion-free bilinear equation. The search led to an investigation of explosion and the effect of point barriers for scalar stochastic differential equations with linear drift; this is used to explain the clustering effect in the usual superprocess.     

A complementary theory of light scattering by homogeneous spheres

A theory of the scattering of electromagnetic waves by homogeneous spheres, the so-called Mie theory, is presented in a unique and coherent manner in this paper. We begin with Maxwell's equations, from which the vector wave equations are derived and solved by means of the two orthogonal solutions to the scalar wave equation. The transverse incident electric field is mapped in spherical coordinates and expanded in known mathematical functions satisfying the scalar wave equation. Determination of the unknown coefficients in the scattered and internal fields is achieved by matching the electromagnetic boundary conditions on the surface of a sphere. Far-field solutions for the electric field are then given in terms of the scattering functions. Transformation of the electric field to the reference plane containing incident and scattered waves is carried out. Extinction parameters and the phase matrix are derived from the electric field perpendicular and parallel to the reference plane. On the basis of the independent-scattering assumption, the theory is extended to cases involving a sample of homogeneous spheres.     

复Banach空间中C-R方程的全纯解

二重复数是复数的一种推广，在其上的全纯映照族对应于Ｃ２上满足复Ｃａｕｃｈｙ－Ｒｉｅｍａｎｎ方程的全纯映照族．可以证明，这样的映照族本质上是由二个单复变数的全纯函数的直乘积所组成的族．本文证明：即使在Ｂａｎａｃｈ空间中，Ｃａｕｃｈｙ－Ｒｉｅｍａｎｎ方程的全纯解，具有同样的性质．     

Light scattering by a homogeneous sphere near a plane boundary

The scattering of electromagnetic waves by a homogeneous sphere near a plane boundary is presented in this paper. The vector wave equations derived from Maxwell’s equations are solved by means of the two orthogonal solutions to the scalar wave equation. Hankel transformation and Erdélyi’s formula are used to satisfy the planar boundary conditions and the determination of the unknown coefficients in the scattered field and internal fields is achieved by matching the electromagnetic boundary conditions on the surface of the sphere. Existence and uniqueness of the solution of the series involving these unknown coefficients are shown.     

Dynamics of a rotating layer of an ideal electrically conducting incompressible inhomogeneous fluid in an equatorial region

The equations describing the three-dimensional equatorial dynamics of an ideal electrically conducting inhomogeneous rotating fluid are studied. The magnetic and velocity fields are represented as superpositions of unperturbed steady-state fields and those induced by wave motion. As a result, after introducing two auxiliary functions, the equations are reduced to a special scalar one. Based on the study of this equation, the solvability of initial-boundary value problems arising in the theory of waves propagating in a spherical layer of an electrically conducting density-inhomogeneous rotating fluid in an equatorial zone is analyzed. Particular solutions of the scalar equation are constructed that describe small-amplitude wave propagation.     

Differentialoperatoren bei partiellen Differentialgleichungen

Summary In the present paper those formally hyperbolic differential equations are characterized for which solutions can be represented by means of differential operators acting on holomorphic functions. This is done by a necessary and sufficient condition on the coefficients of the differential equation. These operators are determined simultaneously. By it a general procedure is presented to construct differential equations and corresponding differential operators which map holomorphic functions onto solutions of the differential equations. We also discuss the question under which circumstances all the solutions of a differential equation can be represented by differential operators. For the equations characterized previously we determine the Riemann function. Some special classes of differential equations are investigated in detail. Furthermore the possibility of a representation of pseudoanalytic functions and the corresponding Vekua resolvents by differential operators is discussed.

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Herrn Prof. Dr. K. W. Bauer zum 60. Geburtstag gewidmet     

Waves produced by sources distributed on an expanding disk

Explicit solutions of a nonhomogeneous wave equation are constructed. The solutions obtained describe waves generated by sources distributed on a disk that expands with velocities less than, equal to, or greater than the velocity of the wavefront. The structure and the directional transmission of waves are discussed. It is shown that the scalar solutions obtained can be applied to electromagnetic waves. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 38–47.     

New exact solutions of the massive Dirac equation with electric or scalar potential

A new class of exact solutions for the massive Dirac equation with electric or scalar potential depending on one independent variable is constructed. This construction is based on quaternionic reformulation of the Dirac equation. Copyright © 2000 John Wiley & Sons, Ltd.     

FRACTIONAL DIFFUSION EQUATIONS WITH INTERNAL DEGREES OF FREEDOM

We present a generalization of the linear one-dimensional diffusion equation by com-bining the fractional derivatives and the internal degrees of freedom. The solutions areconstructed from those of the scalar fractional diffusion equation. We analyze the in-terpolation between the standard diffusion and wave equations defined by the fractionalderivatives. Our main result is that we can define a diffusion process depending on theinternal degrees of freedom associated to the system.     

Special vekua equations with singular coefficients

Special equations of Vekua-type with singular coefficients are considered. As a first step we study the influence of the coefficients of model equations on the choice of the function spaces for its solutions and on the boundary conditions. As an application we sketch the consideration of boundary value problems for Vekua equations with variable coefficients having a strong singularity at z =0     

Nonstationary Electromagnetic Gaussian Beams in Inhomogeneous Anisotropic Media

Nonstationary Gaussian beams of quasiphoton type for the Maxwell equation with an arbitrary anisotropy are constructed. The solutions of the Maxwell equations can be described as ray-type solutions with complex phases and amplitudes. Owing to a large parameter p, they are concentrated in small neighborhoods of space-time rays corresponding to different types of electromagnetic waves in an anisotropic medium. Bibliography: 6 titles.     

A class of explicitly resolvable evolution equations

We consider a class of evolution equations with “scalar nonlinearities” and the associated steady equations. An explicit representation of solutions is obtained in terms of the solution of a scalar nonlinear functional differential equation. Convergence to an equilibrium solution is discussed.     

EVANS FUNCTIONS AND INSTABILITY OF A STANDING PULSE SOLUTION OF A NONLINEAR SYSTEM OF REACTION DIFFUSION EQUATIONS

In this paper, we consider a nonlinear system of reaction diffusion equations arising from mathematical neuroscience and two nonlinear scalar reaction diffusion equations under some assumptions on their coefficients. The main purpose is to couple together linearized stability criterion (the equivalence of the nonlinear stability, the linear stability and the spectral stability of the standing pulse solutions) and Evans functions to accomplish the existence and instability of standing pulse solutions of the nonlinear system of reaction diffusion equations and the nonlinear scalar reaction diffusion equations. The Evans functions for the standing pulse solutions are constructed explicitly.