Riemann Equations Research Articles

Riemann–Hilbert Problems for Hardy Space of Meta-Analytic Functions on the Unit Disc

The aim of this paper is to study Riemann–Hilbert problems for Hardy space of a class of meta-analytic functions defined on the unit disc. Here, the meta-analytic functions we focus on are null-solutions to a class of polynomially Cauchy–Riemann equations. We first establish decomposition theorems for Hardy space of meta-analytic functions defined on the unit disc, and use them to characterize the boundary behavior of Hardy space of meta-analytic functions defined on the unit disc. Then, we make full use of these decomposition theorems and a transform constructed to solve the Riemann–Hilbert problem for Hardy space of a class of meta-analytic functions in two different cases of the parameter involved, separately. Finally, we give explicit integral expressions of solutions and conditions of solvability, respectively.

Single periodic Riemann problems for null-solutions to polynomially Cauchy–Riemann equations

In this paper, we focus on single periodic Riemann problems for a class of meta-analytic functions, i.e. null-solutions to polynomially Cauchy–Riemann equation. We first establish decomposition theorems for single periodic meta-analytic functions. Then, we give a series expansion of single periodic meta-analytic functions, and derive generalised Liouville theorems for them. Next, we introduce a definition of order for single periodic meta-analytic functions at infinity, and characterise their growth at infinity. Finally, applying the decomposition theorem for single periodic meta-analytic functions, we get explicit expressions of solutions and condition of solvability to Riemann problems for single periodic meta-analytic functions with a finite order at infinity.

Riemann–Hilbert-type boundary value problems on a half hexagon

In this article we discuss the explicit solvability of both Schwarz boundary value problem and Riemann–Hilbert boundary value problem on a half hexagon in the complex plane. Schwarz-type and Pompeiu-type integrals are obtained. The boundary behavior of these operators is discussed. Finally, we investigate the Schwarz problem and the Riemann–Hilbert problem for inhomogeneous Cauchy–Riemann equations.

Riemann–Hilbert Problems for Monogenic Functions on Upper Half Ball of $${\mathbb {R}}^4$$ R 4

In this paper we are interested in finding solutions to Riemann–Hilbert boundary value problems, for short Riemann–Hilbert problems, with variable coefficients in the case of axially monogenic functions defined over the upper half unit ball centred at the origin in four-dimensional Euclidean space. Our main idea is to transfer Riemann–Hilbert problems for axially monogenic functions defined over the upper half unit ball centred at the origin of four-dimensional Euclidean spaces into Riemann–Hilbert problems for analytic functions defined over the upper half unit disk of the complex plane. Furthermore, we extend our results to axially symmetric null-solutions of perturbed generalized Cauchy–Riemann equations.

On the origins of intersecting potential energy surfaces

Techniques developed for investigating nonadiabatic processes in molecular systems are adapted to study the structure and properties of holomorphic and meromorphic functions of a complex variable, $$f(z)=\mathfrak {R}(f)+i\,\mathfrak {I}(f)$$ . The connection is that $$\mathfrak {R}(f)$$ and $$\mathfrak {I}(f)$$ are correlated two-dimensional scalar functions, interrelated by the Cauchy–Riemann equations. Exploiting this fact, it is demonstrated that $$\mathfrak {R}(f)$$ and $$\mathfrak {I}(f)$$ of f can be envisaged in Euclidean $${\mathbb {R}}^{3}$$ space as a two-state set of constrained, intersecting two-dimensional potential energy surfaces (PESs), called the graph of f. Importantly, the analytic and algebraic properties of f dictate the geometric structure evinced in the graph of f. This parallels multi-state sets of higher-dimensional, constrained, intersecting PESs linked with correlated electronic eigenstates of the parameterized molecular Hamiltonian operator. In view of this association, the language and mathematical infrastructure devised by chemists for discussing and analyzing intersections in higher-dimensional PESs are suitably modified for f. Notably, an algorithm capable of optimizing roots and poles of f through analysis of the real, two-dimensional $$\mathfrak {R}(f)$$ and $$\mathfrak {I}(f)$$ functions is derived, which is based on intersection-adapted coordinate and constrained Lagrangian methodologies. As constrained, intersecting PESs are indispensible for conceptualizing and characterizing the physics governing nonadiabatic phenomena, f represents a foundational bridge to these more abstract constructions.

On the weighted L2 estimate for the k-Cauchy–Fueter operator and the weighted k-Bergman kernel

The k-Cauchy–Fueter operators, k=0,1,…, are quaternionic counterparts of the Cauchy–Riemann operator in the theory of several complex variables. The weighted L2 method to solve Cauchy–Riemann equation is applied to find the canonical solution to the non-homogeneous k-Cauchy–Fueter equation in a weighted L2-space, by establishing the weighted L2 estimate. The weighted k-Bergman space is the space of weighted L2 integrable functions annihilated by the k-Cauchy–Fueter operator, as the counterpart of the Fock space of weighted L2-holomorphic functions on Cn. We introduce the k-Bergman orthogonal projection to this closed subspace, which can be nicely expressed in terms of the canonical solution operator, and its matrix kernel function. We also find the asymptotic decay for this matrix kernel function.

用于解析函数复分析的共轭边界元法

An analytic function is composed of 2 real conjugate harmonic functions, of which the complex analysis plays an important role in the fields of applied mathematics and mechanics. A set of weighted residual equations were proposed and proved to be equivalent to the approximate solution to the original problem involving 2 governing equations in the domain, the boundary condition and the CauchyRiemann equation at the boundary. 2 conventional direct boundary integral equations at the boundary collocation points were deduced from 2 of the weighted residual equations, and 1 finite difference equation was deduced from the rest one. The mathematical problem arising from the illconditioned linear equations was solved and the Cauchy integral equation was adopted for numerical calculation of the fields at the internal points inside the domain. Finally, the proposed conjugate boundary element method with constant elements was completely established. 3 examples demonstrate that, the proposed method is valid for analytic functions in terms of the power function, the exponential function and the logarithmic function in interior or exterior domains, and the error estimation of the proposed method is at the same order as that of the boundary element method for 2D potential problems.

Solvability of the inhomogeneous Cauchy–Riemann equation in projective weighted spaces

We establish an analog of Hormander’s Theorem on solvability of the inhomogeneous Cauchy–Riemann equation for a space of measurable functions satisfying a system of uniform estimates. The result is formulated in terms of the weight sequence defining the space. The same conditions guarantee the weak reducibility of the corresponding space of entire functions. Basing on these results, we solve the problem of describing the multipliers in weighted spaces of entire functions with the projective and inductive-projective topological structure. Applications are obtained to convolution operators in the spaces of ultradifferentiable functions of Roumieu type.

Tangential Cauchy–Riemann Equations on Pseudoconvex Boundaries of Finite and Infinite Type in $$\mathbb {C}^2$$ C 2

In this paper, we will present what we think is a natural extension of the finite type condition in the sense of Range on pseudoconvex domains. This type condition generalizes the notion of finite type in the original theory as well as consists many cases of infinite type in the sense of Range. Following the integral representation by Henkin, we also provide \(L^p\) and “new” Holder estimates for solutions of \(\bar{\partial }_b\)-equations on the boundaries of domains which emphasize the infinite type condition.

Harmonic wave radiation induced by a flat piston with allowance for the mobility of the interface and nonlinearity of medium

The problem of harmonic pressure wave generation by a moving piston is solved for the first time. An initial boundary value problem for the Riemann equation is formulated, and a boundary condition for the current position of a contact boundary is set. Physical effects caused by the allowance for mobility of the contact boundary and nonlinearity of the medium are considered in the framework of the obtained analytical solution.

Boundary value problem for a generalized Cauchy–Riemann equation with singular coefficients

For a generalized Cauchy–Riemann system whose coefficients admit higher-order singularities on a segment, we obtain an integral representation of the general solution and study a boundary value problem combining the properties of the linear conjugation problem and the Riemann–Hilbert problem in function theory.

A Hilbert Boundary Value Problem for Generalised Cauchy–Riemann Equations

We elaborate a boundary Fourier method for studying an analogue of the Hilbert problem for analytic functions within the framework of generalised Cauchy–Riemann equations. The boundary value problem need not satisfy the Shapiro–Lopatinskij condition and so it fails to be Fredholm in Sobolev spaces. We show a solvability condition of the Hilbert problem, which looks like those for ill-posed problems, and construct an explicit formula for approximate solutions.

New extension phenomena for solutions of tangential Cauchy–Riemann equations

ABSTRACTIn our recent work, we showed that C∞ CR-diffeomorphisms of real-analytic Levi-nonflat hypersurfaces in ℂ2 are not analytic in general. This result raised again the question on the nature of CR-maps of a real-analytic hypersurfaces.In this paper, we give a complete picture of what CR-maps actually are. First, we discover an analytic continuation phenomenon for CR-diffeomorphisms which we call the sectorial analyticity property. It appears to be the optimal regularity property for CR-diffeomorphisms in general. We emphasize that such type of extension never appeared previously in the literature. Second, we introduce the class of Fuchsian type hypersurfaces and prove that (infinitesimal generators of) CR-automorphisms of a Fuchsian type hypersurface are still analytic. In particular, this solves a problem formulated earlier by Shafikov and the first author.Finally, we prove a regularity result for formal CR-automorphisms of Fuchsian type hypersurfaces.

The Poincaré–Bendixson Theorem and the non-linear Cauchy–Riemann equations

Fiedler and Mallet-Paret (1989) prove a version of the classical Poincaré–Bendixson Theorem for scalar parabolic equations. We prove that a similar result holds for bounded solutions of the non-linear Cauchy–Riemann equations. The latter is an application of an abstract theorem for flows with a(n) (unbounded) discrete Lyapunov function.

A global domain/boundary integral equation method for the inverse wave source and backward wave problems

A g-analytic function theory, the Cauchy–Riemann equations of g-analytic function and the g-conformal invariant for the wave equation are derived in this paper. As a consequence, there exist two global spectral relations for the wave equation. By suitably choosing two different types of Trefftz test functions in the derived Green’s second identity, we can recover unknown wave source function very well using the global domain/boundary integral equation method (BIEM), which is robust against large noise up to . Then, we develop a numerical algorithm based on the BIEM, which is effective for the backward wave problem, as well as for the long-term solution of the wave equation. Numerical examples are used to demonstrate the efficiency and accuracy of these methods.

Riemann–Hilbert problems for null-solutions to iterated generalized Cauchy–Riemann equations in axially symmetric domains

We consider Riemann–Hilbert boundary value problems (for short RHBVPs) with variable coefficients for axially symmetric poly-monogenic functions, i.e., null-solutions to iterated generalized Cauchy–Riemann equations, defined in axially symmetric domains. This extends our recent results about RHBVPs with variable coefficients for axially symmetric monogenic functions defined in four-dimensional axially symmetric domains. First, we construct the Almansi-type decomposition theorems for poly-monogenic functions of axial type. Then, making full use of them, we give the integral representation solutions to the RHBVP considered. As a special case, we derive solutions to the corresponding Schwarz problem. Finally, we generalize the result obtained to functions of axial type which are null-solutions to perturbed iterated generalized Cauchy–Riemann equations Dαkϕ=0,k⩾2(k∈N),α∈R.

Schwarz boundary value problem in half lens and half lune

In this article, firstly we obtain Cauchy–Schwarz representation in a half lens in . Then, we consider the Schwarz boundary value problem for the inhomogeneous Cauchy–Riemann equation and explicit expression of solution is obtained. Secondly, the Schwarz problem is explicitly solved in two half lunes which are complements of the half lens with respect to two orthogonal half circles.

“Riemann equations” in bidifferential calculus

We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we obtain a variety of equations, including a semi-discrete and a fully discrete version of the matrix Riemann equation. A corresponding universal solution-generating method then either yields a (continuous or discrete) Cole-Hopf transformation, or leaves us with the problem of solving Riemann equations (hence an application of the hodograph method). If the bidifferential calculus extends to second order, solutions of a system of “Riemann equations” are also solutions of an equation that arises, on the universal level of bidifferential calculus, as an integrability condition. Depending on the choice of bidifferential calculus, the latter can represent a number of prominent integrable equations, like self-dual Yang-Mills, as well as matrix versions of the two-dimensional Toda lattice, Hirota’s bilinear difference equation, (2+1)-dimensional Nonlinear Schrödinger (NLS), Kadomtsev-Petviashvili (KP) equation, and Davey-Stewartson equations. For all of them, a recent (non-isospectral) binary Darboux transformation result in bidifferential calculus applies, which can be specialized to generate solutions of the associated “Riemann equations.” For the latter, we clarify the relation between these specialized binary Darboux transformations and the aforementioned solution-generating method. From (arbitrary size) matrix versions of the “Riemann equations” associated with an integrable equation, possessing a bidifferential calculus formulation, multi-soliton-type solutions of the latter can be generated. This includes “breaking” multi-soliton-type solutions of the self-dual Yang-Mills and the (2+1)-dimensional NLS equation, which are parametrized by solutions of Riemann equations.

Solvability of inhomogeneous Cauchy–Riemann equation in spaces of functions with a system of uniform weight estimates

We obtain an analog of the Hormander theoremon solvability of the \(\overline \partial \)-problemin spaces of functions satisfying a system of uniform estimates. The result is formulated in terms of the weight sequence determining the space. We apply the results for multipliers of projective and inductiveprojective weight spaces of entire functions and for convolution operators in the Roumieu spaces of ultradifferentiable functions.

Hidden gauge symmetry in holomorphic models

We study the effect of a hidden gauge symmetry on complex holomorphic systems. For this purpose, we show that intrinsically any holomorphic system has this gauge symmetry. We establish that this symmetry is related to the Cauchy–Riemann equations, in the sense that the associated constraint is a first class constraint only in the case that the potential be holomorphic. As a consequence of this gauge symmetry on the complex space, we can fix the gauge condition in several ways and project from the complex phase-space to real phase space. Different projections are gauge related on the complex phase-space but are not directly related on the real physical phase-space.