Solution Of Riemann-Hilbert Problem Research Articles

Riemann–Hilbert problem for the focusing nonlinear Schrödinger equation with multiple high-order poles under nonzero boundary conditions

The Riemann–Hilbert (RH) problem is developed to study the focusing nonlinear Schrödinger (NLS) equation with multiple high-order poles under nonzero boundary conditions. Laurent expansion and Taylor series are employed to replace the residues at the simple- and the second-poles. Furthermore, the solution of RH problem is transformed into a closed system of algebraic equations, and the soliton solutions corresponding to the transmission coefficient 1/s11(z) with an N-order pole are obtained by solving the algebraic system. Then, in a more general case, the transmission coefficient with multiple high-order poles is studied, and the corresponding solutions are obtained. In addition, for high-order pole, the propagation behavior of the soliton solution corresponding to a third-order pole and the mixed case of a second-order pole and a simple pole are given as example.

ELLIPTIC ASYMPTOTIC REPRESENTATION OF THE FIFTH PAINLEVÉ TRANSCENDENTS

For the fifth Painlevé transcendents, an asymptotic representation by the Jacobi sn-function is presented in cheese-like strips along generic directions near the point at infinity. Its elliptic main part depends on a single integration constant, which is the phase shift and is parametrized by monodromy data for the associated isomonodromy deformation. In addition, under a certain supposition, the error term is also expressed by an explicit asymptotic formula, whose leading term is written in terms of integrals of the sn-function and the ϑ-function, and contains the other integration constant. Instead of the justification scheme for asymptotic solutions of Riemann-Hilbert problems by the Brouwer fixed point theorem, we begin with a boundedness property of a Lagrangian function, which enables us to determine the modulus of the sn-function satisfying the Boutroux equations and to construct deductively the elliptic representation.

Improving radiation performance of the cylindrical dielectric reflector sandwiched by thin resistive layer illuminated by a complex line source

We studied two-dimensional (2-D) thin dielectric parabolic reflector, sandwiched by thin resistive layer from both sides. It is illuminated by an E-polarized electromagnetic plane wave from front side. It is expected to observe the radiation performance of such a composite reflector depending on the electrical and geometrical parameters. We applied two-sided generalized boundary conditions which are modified for a thin multilayer structure. The electromagnetic boundary value problem is formulated and it finally produced a set of two coupled singular integral equations (SIEs). They are handled with the analytical regularization method based on the Riemann-Hilbert Problem solution. The final numerical results are in the Fredholm second-kind matrix equation form and any desired accuracy can possibly be obtained. Also the solution procedure guarantees the convergence. It is observed that the proper selection of the electrical resistivity of the thin resistive layer produces wider bandwidth in the aperture efficiency and the directivity variation approaches to the perfect electric conductor (PEC) reflector case for higher dielectric constants. This selection can increase the allowable frequency band in the applications by coating dielectric layer from both sides. We demonstrate these statements in the numerical results section in the microwave frequency range.

Evaluation of the E‐polarization focusing ability in Thz range for microsize cylindrical parabolic reflector made of thin dielectric layer sandwiched between graphene

We consider two-dimensional (2-D) thin dielectric parabolic reflector, covered with graphene from both sides, illuminated symmetrically by an E-polarized electromagnetic plane wave. Our aim is to estimate the focussing ability of such a composite reflector depending on the graphene parameters. We use a version of the two-side generalized boundary condition, modified for a thin multilayer case. The scattering is formulated as an electromagnetic boundary-value problem; it is cast to a set of two coupled singular integral equations that are further subjected to analytical regularisation based on the known Riemann–Hilbert problem solution. Thanks to this procedure, the numerical results are computed from a Fredholm second-kind matrix equation that guarantees convergence and provides easily controlled accuracy. In the lower part of the THz range, high values of the focusing ability are observed even for a thin reflector; they are greater than for a purely dielectric reflector and a free standing graphene reflector. On the other hand, a regime of almost full transparency, intrinsic for the dielectric layer, can spoil focusing ability. Novel aspect is that the location in frequency of this effect can be controlled, in wide range, by changing the chemical potential of graphene.

Electromagnetic characterization of tuneable graphene‐strips‐on‐substrate metasurface over entire THz range: Analytical regularization and natural‐mode resonance interplay

Scattering and absorption of the H-polarized plane wave by the infinite grating of flat graphene strips are considered in the environment met most frequently—on or at the surface of a dielectric-slab substrate. The full-wave meshless code is based on the analytical semi-inversion using the Riemann–Hilbert problem solution. This leads to a Fredholm second-kind matrix equation for the Floquet harmonic amplitudes that guarantees code convergence and provides easy control of computational error, which can be reduced to machine precision. The matrix elements are combinations of elementary functions, and therefore, the code is accurate and quite economical. This enables computation of the reflectance, transmittance, and absorbance as a function of the frequency in the wide band from static case to 10 THz. Numerical results show that such a metasurface with micrometre-sized strips is a composite periodic open resonator. It is highly frequency-selective thanks to the interplay of three types of natural modes—low-Q slab, moderate-Q plasmon strip, and ultra-high-Q lattice—that do not exist in the absence of the substrate. Varying the chemical potential of graphene, one can manipulate the electromagnetic characteristics of the metasurface at a fixed frequency from almost total transmission to almost total reflection.

Soliton resolution for the short-pulse equation

In this paper, we apply ∂‾ steepest descent method to study the Cauchy problem for the nonlinear short-pulse equationuxt=u+16(u3)xx,u(x,0)=u0(x)∈H(R), where H(R)=W3,1(R)∩H2,2(R) is a weighted Sobolev space. The solution of the short-pulse equation is constructed via a solution of Riemann-Hilbert problem in the new scale (y,t). In any fixed space-time coneC(y1,y2,v1,v2)={(y,t)∈R2:y=y0+vt,y0∈[y1,y2], v∈[v1,v2]}, we compute the long time asymptotic expansion of the solution u(x,t), which implies soliton resolution conjecture consisting of three terms: the leading order term can be characterized with an N(I)-soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the cone; the second |t|−1/2 order term coming from soliton-radiation interactions on continuous spectrum up to an residual error order O(|t|−1) from a ∂‾ equation. Our results also show that soliton solutions of the short-pulse equation are asymptotically stable.

Full-Spectrum Periodic Nonlinear Fourier Transform Optical Communication Through Solving the Riemann-Hilbert Problem

In this article, for the first time, a full-spectrum periodic nonlinear Fourier transform (NFT)-based communication system with the inverse transformation at the transmitter performed by using the solution of Riemann-Hilbert problem (RHP), is proposed and studied. The entire control over the nonlinear spectrum rendered by our technique, where we operate with two qualitatively different components of this spectrum represented, correspondingly, in terms of the main spectrum and the phases, allows us to design a time-domain signal tailored to the characteristics of the transmission channel. In the heart of our system is the RHP-based signal processing utilised to generate the time-domain signal from the modulated nonlinear spectrum. This type of NFT processing leads to a computational complexity that scales linearly with the number of time-domain samples, and we can process signal samples in parallel. In this article, we suggest the way of getting an exactly periodic signal through the correctly formulated RHP, and present evidence of the analogy between band-limited (in ordinary Fourier sense) signals and finite-band (in RHP sense) signals. Also, for the first time, we explain how to modulate the phases of individual periodic nonlinear modes. The performance of our transmission system is evaluated through numerical simulations in terms of bit error rate and Q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -factor dependencies on the transmission distance and power, and the results demonstrate the good potential of the approach.

H-polarized plane-wave scattering by a PEC strip grating on top of a dielectric substrate: analytical regularization based on the Riemann-Hilbert Problem solution

ABSTRACTWe consider the H-polarized plane wave scattering from an infinite flat grating of perfectly electrically conducting strips, placed on the interface of a dielectric slab. We reduce this problem to a dual series equation for the complex amplitudes of the Floquet spatial harmonics. Then, we perform analytical regularization of this equation, based on the inversion of the static part of the problem with the aid of the Riemann-Hilbert Problem. This yields a Fredholm second-kind infinite matrix equation, numerical solution of which has a guaranteed convergence. Numerical results obtained demonstrate how the rate of convergence depends on the geometrical parameters and then concentrate on the resonance effects in the reflection and transmission. We reveal and discuss ultra-high-Q resonances on the lattice modes of such a composite grating, overlooked in earlier studies.

Solving Riemann-Hilbert problems with meromorphic functions

Abstract In this paper, we introduce the use of a powerful tool from theoretical complex analysis, the Blaschke product, for the solution of Riemann-Hilbert problems. Classically, Riemann-Hilbert problems are considered for analytic functions. We give a factorization theorem for meromorphic functions over simply connected nonempty proper open subsets of the complex plane and use this theorem to solve Riemann-Hilbert problems where the given data consists of a meromorphic function.

TBA equations and resurgent Quantum Mechanics

We derive a system of TBA equations governing the exact WKB periods in one-dimensional Quantum Mechanics with arbitrary polynomial potentials. These equations provide a generalization of the ODE/IM correspondence, and they can be regarded as the solution of a Riemann-Hilbert problem in resurgent Quantum Mechanics formulated by Voros. Our derivation builds upon the solution of similar Riemann-Hilbert problems in the study of BPS spectra in mathcal{N} = 2 gauge theories and of minimal surfaces in AdS. We also show that our TBA equations, combined with exact quantization conditions, provide a powerful method to solve spectral problems in Quantum Mechanics. We illustrate our general analysis with a detailed study of PT-symmetric cubic oscillators and quartic oscillators.

On the Riemann-Hilbert factorization problem for positive definite functions

We give several general theorems concerning positive definite solutions of Riemann-Hilbert problems on the real line. Furthermore, as an example, we apply our theory to the characteristic function of a class of Levy processes and we find the distribution of their extrema at a given stopping time.

Asymptotic for a Riemann-Hilbert problem solution

We solve an homogeneous Riemann-Hilbert problem, such that the density does not satisfy the zero index condition. Moreover, we find an asymptotic representation for the solution to this problem. As an application, the problem mentioned above appears in the construction of a Green operator for a linear partial differential equation with Riesz fractional derivative. Also, in order to estimate such operator the asymptotic representation is used.

Fuchsian Riemann-Hilbert problem for “real doubles” of Hurwitz Frobenius manifolds

A solution to the matrix Fuchsian Riemann-Hilbert problem (inverse monodromy problem) corresponding to “real doubles” of Dubrovin's Frobenius structures on Hurwitz spaces is constructed. The solution is given in terms of certain meromorphic differentials integrated over a basis of an appropriate relative homology space of the Riemann surface. The relationship with the solution of Fuchsian Riemann-Hilbert problem for Dubrovin's Hurwitz Frobenius manifolds is established. A solution of the Riemann-Hilbert problem corresponding to deformations of the “real doubles” is also given.

Riemann–Hilbert formulation for the KdV equation on a finite interval

The initial-boundary value problem for the KdV equation on a finite interval is analyzed in terms of a singular Riemann–Hilbert problem for a matrix-valued function in the complex k-plane which depends explicitly on the space–time variables. For an appropriate set of initial and boundary data, we derive the k-dependent “spectral functions” which guarantee the uniqueness of Riemann–Hilbert problem's solution. The latter determines a solution of the initial-boundary value problem for KdV equation, for which an integral representation is given. To cite this article: I. Hitzazis, D. Tsoubelis, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

On Monodromy of Generalized Analytic Functions

We discuss a number of topics concerned with certain boundary-value problems in the context of generalized analytic functions. Solution of the classical Riemann-Hilbert problem and the linear conjugation problem for analytic functions is described in appropriate function classes and the same scheme is applied to generalized analytic functions and vectors. In particular, we describe solution of the Riemann-Hilbert problem for generalized analytic functions and obtain an explicit analytic presentation of monodromy matrices in the case of generalized analytic vectors.

A priori Lp-estimates for solutions of Riemann-Hilbert problems

We use steepest-descent ideas to obtain a priori Lp-estimates for solutions of Riemann-Hilbert problems. Such estimates play a crucial role, in particular, in analyzing the long-time behavior of solutions of the perturbed nonlinear Schrödinger equation on the line.

Wiener-Hopf factorization for a class of oscillatory symbols

Two classes of 2×2 matrix symbols involving oscillatory functions are considered, one of which consists of triangular matrices. An equivalence theorem is obtained, concerning the solution of Riemann-Hilbert problems associated with each of them. Conditions for existence of canonical generalized factorization are established, as well as boundedness conditions for the factors. Explicit formulas are derived for the factors, showing in particular that only one of the columns needs to be calculated. The results are applied to solving a corona problem.

Explicit solution of Riemann-Hilbert problems for the Ernst equation

Riemann\char21{}Hilbert problems are an important solution technique for completely integrable differential equations. They are used to introduce a free function in the solutions which can be used at least in principle to solve initial or boundary value problems. But even if the initial or boundary data can be translated into a Riemann\char21{}Hilbert problem, it is in general impossible to obtain explicit solutions. In the case of the Ernst equation, however, this is possible for a large class because the matrix problem can be shown to be gauge equivalent to a scalar one on a hyperelliptic Riemann surface that can be solved in terms of theta functions. As an example we discuss the rigidly rotating dust disk.

The effect of internal pressure on a penny-shaped crack at the interface of two bonded dissimilar micropolar elastic half-spaces

In the linear theory of micropolar elasticity, the problem of a penny-shaped crack at the interface of two bonded dissimilar micropolar elastic half spaces is studied. The problem is first reduced to a system of dual integral equations which are further reduced to the solution of Riemann-Hilbert problem. Further stresses at the rims of cracks and in the vicinity have been evaluated.

Numerical conformal mapping methods based on function conjugation

A unifying treatment of methods for computing conformal maps from the unit disk onto a Jordan region is presented. Integral and integro-differential equations (involving the conjugation operator) for the boundary correspondence function are first derived using an arbitrary auxiliary function having certain properties. Then various iteration methods for solving these equations are described in this generality, so that the basic ideas become manifest. Specific methods are then treated as examples of the general theory. Among them are, in particular, the successive conjugation methods of Theodorsen, Melentiev and Kulisch, Timman, and Friberg, the projection method of Bergström, and the Newton methods of Vertgeim, Wegmann, and Hübner (which make use of the easy construction of the solutions of Riemann-Hilbert problems). Many of these methods are treated in greater generality than in the literature. The connections with the methods of Fornberg, Menikoff-Zemach, Chakravarthy-Anderson, and Challis-Burley are also outlined.