Homogeneous Hilbert Problem Research Articles

A homogeneous Hilbert problem with discontinuous coefficients and two-side curling at infinity point of order 1/2 ≤ ρ < 1

We study a homogeneous Riemann—Hilbert boundary-value problem in the upper half of the complex plane with a countable set of coefficient discontinuities and two-side curling at infinity. We obtain a general solution in the case when the problem index has a power singularity of order ρ, 1/2 ≤ ρ < 1, and study the solvability conditions.

To the Solution of the Hilbert Problem with Infinite Index

In this paper, we obtain a generalization of the method of regularizing multipliers for the solution of the Hilbert boundary-value problem with finite index in the theory of analytic functions to the case of an infinite power-behaved index. This method is used to obtain a general solution of the homogeneous Hilbert problem for the half-plane, a solution that depends on the existence and the number of entire functions possessing mirror symmetry with respect to the real axis and satisfying some additional constraints related to the singularity characteristic of the index. To solve of the inhomogeneous problem, we essentially use a specially constructed solution of the homogeneous problem whereby we reduce the boundary condition of the Hilbert problem to a Dirichlet problem.

Review: Aspects of Solution-generating Techniques for Space-times with Two Commuting Killing Vectors

Certain aspects of solution-generatingtechniques for spacetimes with two commuting Killingvectors are reviewed. A brief historical introduction tostationary axisymmetric systems is given. The importance of the Homogeneous Hilbert problem associatedwith the equations, unifying the group-theoretic withthe soliton-theoretic approaches, is emphasized. Theformalism of generating functions is introduced, both for vacuum and electrovacuum.Sibgatullin's technique for electrovacuum solutions isrelated to the Hauser Ernst variables and a method byErnst is briefly discussed. The solitonic methods ofBelinsky-Zakharov and Alekseev are reviewed. Their relation isemphasized by an explicit proof, at the level ofgenerating techniques, that the BZ two soliton with twocomplex conjugate poles is isomorphic to the Alekseev one-soliton (restricted to vacuum) with trivialgauge. The Alekseev non-soliton technique is discussed.Some recent developments are brieflydiscussed.

Green's function for generalized Hilbert problem for cracks and free surfaces in composite materials

Green's function for a generalized vector Hilbert problem is calculated which can be used to solve the Hilbert problem with any integrable inhomogeneity. The Green's function is obtained by using a complex transform defined by eigenfunctions of the homogeneous Hilbert problem. This method should be particularly convenient for the stress analysis of anisotropic composite materials containing cracks and free surfaces. The method is illustrated by applying it to calculate the stress field in an anisotropic bimaterial composite containing an interfacial crack. It is found that, in agreement with the earlier published work on isotropic composites, the stress field is oscillatory but, except very near the crack tip, the oscillations are negligible. Numerical results are presented for a stress field in a uniformly loaded (shear and compressive) Cu/Ni layered composite containing an interfacial crack as well as cracked homogeneous copper.

Initial value problem for colliding gravitational plane waves. IV

In a preceding paper [I. Hauser and F. J. Ernst, J. Math. Phys. 31, 871 (1990)], the initial value problem for colliding gravitational plane waves was reformulated as a homogeneous Hilbert problem (HHP) in a complex plane. It is now proven that a unique solution of this HHP always exists and that any assumed differentiability or holomorphy properties of the plane-wave metrics imply corresponding properties for the scattered wave metric. This completes the demonstration, initiated in the preceding paper, that the solution of the HHP furnishes the solution of the initial value problem. A Fredholm equation that is equivalent to the HHP and that was introduced in the preceding paper is used to effect the proofs.

Initial value problem for colliding gravitational plane waves. III

The development of a homogeneous Hilbert problem (HHP) approach to the initial value problem (IVP) for colliding gravitational plane waves with noncollinear polarization that began in two earlier papers [I. Hauser and F. J. Ernst, J. Math. Phys. 30, 872 (1989) and 30, 2322 (1989)] is continued. After formulating the HHP, the description of how one can apply it to generate a new family of solutions of the colliding wave problem that generalizes a three-parameter family constructed by Ernst, García, and Hauser [J. Math. Phys. 29, 681 (1988)] using a double-Harrison transformation is given. Then the proof that the solution of the new HHP indeed solves the IVP that is posed is presented. A matrix Fredholm equation of the second kind that is equivalent to the HHP is also deduced. This will be used in a sequel to complete the proof of existence of solutions of the HHP and the proof that certain assumed differentiability hypotheses are in fact valid.

Colliding gravitational plane waves with noncollinear polarization. III

The Hauser–Ernst homogeneous Hilbert problem (HHP) approach, formerly used in connection with the derivation of stationary axisymmetric fields, is here adapted to the derivation of colliding gravitational plane wave solutions of the vacuum Einstein equations. Proceeding from Kasner metrics, and using a double-Harrison transformation, the HHP approach yields a three-parameter generalization of a two-parameter family of colliding wave solutions found recently by Ferrari, Ibañez, and Bruni. In the present paper we provide the details concerning the derivation of this new family of solutions, and we set the stage for future applications of the HHP approach in connection with gravitational waves.

The application of the homogeneous Hilbert problem of Hauser and Ernst to cosmological models with spatial axes of symmetry

Open and closed vacuum and scalar field/stiff perfect fluid cosmological models requiring more than one canonical coordinate patch to cover the entire spacetime are treated. The boundary conditions which are met on the spatial axes of symmetry and the matching conditions which hold on the null hypersurfaces joining separate canonical coordinate patches are given. It is shown how all such cosmological solutions may be generated from flat space using the homogeneous Hilbert problem of Hauser and Ernst (1981). Specific solutions considered include the Gowdy solutions, the Friedmann-Robertson-Walker (FRW) models, the Kantowski-Sachs universe, locally rotationally symmetric Bianchi type I, II, III, VIII, IX models and Belinskii's solitonic perturbations of the FRW models.

Noniterative method for constructing many-parameter solutions of the Einstein and Einstein–Maxwell field equations

We present a noniterative method of executing a large class of Kinnersley–Chitre transformations in both the vacuum and the electrovac case. By solving the homogeneous Hilbert problem in the Hauser–Ernst formalism, we generate new many-parameter solutions of the Einstein equations. In the vacuum case, the solution is a natural generalization of the N-fold Neugebauer solution, while, in the electrovac case, we have a natural generalization of the N-fold Cosgrove solution worked out by Wang, Guo, and Wu.

Prolongation structure, Bäcklund Transformation and Principal Homogeneous Hilbert Problem in General Relativity††The invited talk given by one of us (GUO Han-ying) at the third Marcel Grossmann Meeting, Aug. 30-Sept. 3 1982, Shanghai, is mainly abstracted from this paper.

We will briefly review the prolongation structure theory for nonlinear systems, such as nonlinear evolution equations and soliton equations, and its applications to the solution generating techniques for the gravitational fields with two commuting Killing vectors.

Prolongation Structure Approach to the Group-Theoretical Technique for Generating Stationary Axisymmetric Space-Times

The group-theoretical technique for generating stationary axisymmetric gravitational fields is approached by means of the prolongation structure theory for soliton systems. An sp(2)× structure is obtained via solving the fundamental equation for prolongation structures and the F-equation for Kinnersley-Chitre's generating function is naturally introduced as an inverse scattering equation. A homogeneous Hilbert problem (HHP) associated with the Geroch group K and a corresponding linear singular integral equation are derived based upon a general condition satisfied by the auto-Bäcklund transformations in the sense of prolongation structure theory.

On the Prolongation Structure of Ernst Equation

An SL(2R) × R1(l) prolongation structure of Ernst equation with a real parameter l and the corresponding Riccati equation as well as a pair of linear equations which are in principle equivalent to the inverse scattering problem due to Belinsky and Zakharov are obtained by solving the fundamental equation for the prolongation structure. A necessary condition which should be satisfied by the Bäcklund transformations is presented in terms of prolongation structure. And it is indicated that in the, case of Ernst equation the Harrison transformation, Neuge-bauer transformations and other available Bäcklund transformations as well as Belinsky-Zakharov's Riemann transformation, i.e., the homogeneous Hilbert problem (HHP), would be covered by this condition.

Relationship between the inverse scattering techniques of Belinskii–Zakharov and Hauser–Ernst in general relativity

We make a quantitative comparison between the pure-nonsoliton part of the inverse scattering method of Belinskii and Zakharov (BZ) and the homogeneous Hilbert problem of Hauser and Ernst (HE), these being two independent representations of an infinite-dimensional subgroup 𝒦 of the Geroch group K of invariance transformations for spacetimes with two commuting Killing vectors. An explicit formula for the BZ representing matrix function G0(λ) in terms of the HE representing matrix function u(t) is derived. It is shown how certain solution generating techniques (e.g., Harrison’s Bäcklund transformation, HKX transformation, generation of Weyl solution from flat space, generation of n-Kerr–NUT solution from n-Schwarzschild) can be derived directly from the BZ formalism, including the soliton part in some cases, thereby bringing our understanding of the BZ formalism up to the level of the more fully developed HE formalism. A technical point which needed to be resolved along the way was how to analytically continue the complex matrix potential F(t) across a quadratic branch cut and onto the second Riemann sheet. Finally, we consider how the subgroup 𝒦⊆K represented by the BZ and HE formalisms can be enlarged either by simple limiting transitions or by relaxing boundary conditions.

A homogeneous Hilbert problem for the Kinnersley–Chitre transformations of electrovac space-times

The homogeneous Hilbert problem which we recently formulated for Kinnersley–Chitre transformations of vacuum spacetimes is here generalized to handle transformations of electrovac spacetimes. This provides in particular a simple derivation of our previously published integral equation.

A homogeneous Hilbert problem for the Kinnersley–Chitre transformations

A homogeneous Hilbert (Riemann) problem (HHP) is introduced for carrying out the Kinnersley–Chitre transformations of the set V of all axially symmetric stationary vacuum spacetimes, and the spacetimes which are like the axially symmetric stationary ones except that both Killing vectors are spacelike. A proof, which is independent of the Kinnersley–Chitre formalism, establishes that the HHP transforms the potential (for certain closed self-dual 2 forms) F0(x, t) of any given member of V into the potential F (x, t) of another member of V. Two illustrative examples involving the Minkowski space F0(x, t) are given. The representation used for the Geroch group K, the singularities and gauge of the potentials, and possible applications of the HHP are discussed.

The fundamental matrix of the Hilbert problem in transport theory

A method of solving the homogeneous Hilbert problem encountered in the two-media case of the one-dimensional transport equation with degenerate scattering kernels is developed. The existence of the corresponding fundamental matrix is proved when the fundamental matrices for both the full-range problems exist and the elements of these matrices and of their inverses belong to a class L/sub p/, p greater than 2, and the zeroes of the dispersion functions are of integer order. Matrices related to the fundamental matrices satisfy a Fredholm equation whose free term involves unknown polynomial elements. Means of constructing the fundamental matrix from certain solutions of this equation are investigated. Sufficient supplementary conditions are presented in the form of an algebraic equation. Since the kernel is simple in form and its norm approaches zero with the cross sections of both media approaching each other, the Fredholm equation is very suitable for computation.

On a singular integral equation in the theory of radiative transfer

A singular integral equation arising in the theory of radiative transfer with polynomial scattering indicatrices is studied in this paper. In the case of a semi-infinite atmosphere, brightness coefficients and auxiliary functions satisfy this equation when the free term is suitably specified. The general solution ψ of the singular equation is given a closed form in terms of its moments ψn and of the fundamental solution of a homogeneous Hilbert problem. The ψn's (n ⩾ 1) are seen to satisfy a set of linear equations, and ψ 0 turns out to be simply related to the subsequent ψn's.

Useful identities for half-space problems in linear transport theory

A typical translationally invariant linear transport problem for a half-space is solved by the eigenfunction method. Using some derived identities for the solution of the associated homogeneous Hilbert problem it is shown that the usual cumbersome integrals occurring in half-space problems can be reduced to a simple form. The usefulness of the identities is illustrated by solving the problem of a plasma confined to a half-space by a diffuse reflecting wall.