The analytic continuation of solutions to elliptic boundary value problems in two independent variables

An explicit representation is derived for the continuation across an analytic boundary of the solution to a boundary value problem for an analytic elliptic equation of second order in two independent variables. The representation is in terms of Cauchy data on the boundary and the complex Riemann function. This is equivalent to a representation for the solution to Cauchy's problem given by Henrici in 1957. It is confirmed that the method of complex characteristics is satisfactory for locating real singularities in the solution provided that the Riemann function is entire in its four arguments. Applications to Laplace's and Helmholtz's equations are discussed. By inserting known, simple solutions to the latter equation into the representation formula, several nontrivial integral relations involving the Bessel function J₀, and a possibly new series expansion for J_μ(x), are found.     

The modified KdV equation on a finite interval

We analyse an initial-boundary value problem for the mKdV equation on a finite interval by expressing the solution in terms of the solution of an associated matrix Riemann–Hilbert problem in the complex k-plane. This Riemann–Hilbert problem has explicit (x,t)-dependence and it involves certain functions of k referred to as “spectral functions”. Some of these functions are defined in terms of the initial condition q(x,0)=q₀(x), while the remaining spectral functions are defined in terms of two sets of boundary values. We show that the spectral functions satisfy an algebraic “global relation” that characterize the boundary values in spectral terms. To cite this article: A. Boutet de Monvel, D. Shepelsky, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

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).     

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.     

On a Non-symmetric Problem in Electrochemical Machining

We consider a free boundary value problem arising from a non-symmetric problem of electrochemical machining (ECM). After a conformal mapping of the unknown domain the problem is transformed to a non-smooth non-linear Riemann–Hilbert problem for holomorphic functions in the complex unit disk. A special technique allows to remove the singularities in the boundary condition. Utilizing existence results for smooth non-orientable Riemann–Hilbert problems, existence and uniqueness of a solution are shown. Finally, we propose an iterative method of Newton type for the effective numerical computation of the free boundary of the anode and present some test results. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Long-time asymptotics of the focusing Kundu–Eckhaus equation with nonzero boundary conditions

The long-time asymptotics of the focusing Kundu–Eckhaus equation with nonzero boundary conditions at infinity is investigated by the nonlinear steepest descent method of Deift and Zhou. Three asymptotic sectors in space–time plane are found: the plane wave sector I, plane wave sector II and an intermediate sector with a modulated one-phase elliptic wave. The asymptotic solutions of the three sectors are proposed by successively deforming the corresponding Riemann–Hilbert problems to solvable model problems. Moreover, a time-dependent g-function mechanism is introduced to remove the exponential growths of the jump matrices in the modulated one-phase elliptic wave sector. Finally, the modulational instability is studied to reveal the criterion for the existence of modulated elliptic waves in the central region.     

The Camassa–Holm equation on the half-line

We study the initial-boundary-value problem for the Camassa–Holm equation on the half-line by associating to it a matrix Riemann–Hilbert problem in the complex k-plane; the jump matrix is determined in terms of the spectral functions corresponding to the initial and boundary values. We prove that if the boundary values $u(0,t)$ are ?0 for all t then the corresponding initial-boundary-value problem has a unique solution, which can be expressed in terms of the solution of the associated RH problem. In the case $u(0,t)<0$, the compatibility of the initial and boundary data is explicitly expressed in terms of an algebraic relation to be satisfied by the spectral functions. To cite this article: A. Boutet de Monvel, D. Shepelsky, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Corner singularities and boundary layers in a simple convection-diffusion problem

A singularly perturbed convection-diffusion problem posed on the unit square is considered. Its solution may have exponential and parabolic boundary layers, and corner singularities may also be present. Pointwise bounds on the solution and its derivatives are derived. The dependence of these bounds on the small diffusion coefficient, on the regularity of the data, and on the compatibility of the data at the corners of the domain are all made explicit. The bounds are derived by decomposing the solution into a sum of solutions of elliptic boundary-value problems posed on half-planes, then analyzing these simpler problems.     

Boundary value problems for two types of degenerate elliptic systems in <Emphasis Type="Italic">R</Emphasis><Superscript>4</Superscript>

Firstly, the Riemann boundary value problem for a kind of degenerate elliptic system of the first order equations in R ⁴ is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system’s solution, the boundary value problem as stated above is transformed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R ⁴ are derived.     

二阶复椭圆Riemann—Hilbert边值问题

讨论了一般二阶非线性椭圆复方程的Riemann-Hilbert边值问题,首先给出Riemann-Hilbert问题及其适应性的概念,其次给出改进后的边值问题解的表述并证明了它的可解性,最后导出原Rremann-Hilbert边值问题的可解条件。     

Bound-state soliton and rogue wave solutions for the sixth-order nonlinear Schrödinger equation via inverse scattering transform method

In this work, inverse scattering transform for the sixth-order nonlinear Schrödinger equation with both zero and nonzero boundary conditions at infinity is given, respectively. For the case of zero boundary conditions, in terms of the Laurent's series and generalization of the residue theorem, the bound-state soliton is derived. For nonzero boundary conditions, using the robust inverse scattering transform, we present a matrix Riemann–Hilbert problem of the sixth-order nonlinear Schrödinger equation. Then, based on the obtained Riemann–Hilbert problem, the rogue wave solutions are derived through a modified Darboux transformation. Besides, according to some appropriate parameters choices, several graphical analysis are provided to discuss the dynamical behaviors of the rogue wave solutions and analyze how the higher-order terms affect the rogue wave.     

A collocation method for solving a class of singular nonlinear two-point boundary value problems

In this paper, an iterative algorithm for solving singular nonlinear two-point boundary value problems is formulated. This method is basically a collocation method for nonlinear second-order two-point boundary value problems with singularities at either one or both of the boundary points. It is proved that the iterative algorithm converges to a smooth approximate solution of the BVP provided the boundary value problem is well posed and the algorithm is applied appropriately. Error estimates for uniform partitions are also investigated. It has been shown that, for sufficiently smooth solutions, the method produces order h⁴ approximations. Numerical examples are provided to show the effectiveness of the algorithm.     

The Riemann Hilbert Problem for General Elliptic Complex Equations of Fourth Order

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Convergence of the finite element method for an elliptic equation with strong degeneration

Under consideration are the questions of the numerical solution by the finite element method (FEM) of the first boundary value problem for an elliptic equation with degeneration on a part of the boundary. The weak and strong variational statements are posed in the function spaces with the coordinated weights that correspond to the problem. Using the method of the multiplicative extraction of singularities for the finite element method that utilizes piecewise linear elements, we prove that the convergence of the approximate solutions to the exact solution in the weighted norm is not worse than in the case of an elliptic equation without degeneration.     

Debonding of an elastic inhomogeneity of arbitrary shape in anti-plane shear

We investigate the anti-plane shear problem of a curvilinear crack lying along the interface of an arbitrarily shaped elastic inhomogeneity embedded in an infinite matrix subjected to uniform stresses at infinity. Complex variable and conformal mapping techniques are used to derive an analytical solution in series form. The problem is first reduced to a non-homogeneous Riemann–Hilbert problem, the solution of which can be obtained by evaluating the associated Cauchy integral. A set of linear algebraic equations is obtained from the compatibility condition imposed on the resulting analytic function defined in the inhomogeneity and its Faber series expansion. Each of the unknown coefficients in the corresponding analytic functions can then be uniquely determined by solving the linear algebraic equations, which are written concisely in matrix form. The resulting analytical solution is then used to quantify the displacement jump across the debonded section of the interface as well as the traction distribution along the bonded section of the interface. In addition, our solution allows us to obtain mode-III stress intensity factors at the two crack tips. The solution to the anti-plane problem of a partially debonded elliptical inhomogeneity containing a confocal crack is also derived using a similar method.     

A CLASS OF COMPOUND VECTOR-VALUED PROBLEM AND FACTORIZATION OF MATRIX FUNCTION

In this article,we consider a class of compound vector-valued problem on upper-half plane C+,which consists of vector Riemann problem along a closed contour in C+ with matrix coefficient in Hlder class and vector Hilbert problem on the real axis with essential bounded measurable matrix coefficient.Under appropriate assumption we obtain its solution by use of Corona theorem and factorization of matrix functions in decomposed Banach algebras.     

Long‐Time Asymptotics for the Focusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions at Infinity and Asymptotic Stage of Modulational Instability

We characterize the long‐time asymptotic behavior of the focusing nonlinear Schrödinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity by using a variant of the recently developed inverse scattering transform (IST) for such problems and by employing the nonlinear steepest‐descent method of Deift and Zhou for oscillatory Riemann‐Hilbert problems. First, we formulate the IST over a single sheet of the complex plane without introducing the uniformization variable that was used by Biondini and Kova?i? in 2014. The solution of the focusing NLS equation with nonzero boundary conditions is thereby associated with a matrix Riemann‐Hilbert problem whose jumps grow exponentially with time for certain portions of the continuous spectrum. This growth is the signature of the well‐known modulational instability within the context of the IST. We then eliminate this growth by performing suitable deformations of the Riemann‐Hilbert problem in the complex spectral plane. The results demonstrate that the solution of the focusing NLS equation with nonzero boundary conditions remains bounded at all times. Moreover, we show that, asymptotically in time, the xt ‐plane decomposes into two types of regions: a left far‐field region and a right far‐field region, where the solution equals the condition at infinity to leading order up to a phase shift, and a central region in which the asymptotic behavior is described by slowly modulated periodic oscillations. Finally, we show how, in the latter region, the modulus of the leading‐order solution, initially obtained as a ratio of Jacobi theta functions, can be reduced to the well‐known elliptic solutions of the focusing NLS equation. These results provide the first characterization of the long‐time behavior of generic perturbations of a constant background in a modulationally unstable medium. © 2017 Wiley Periodicals, Inc.     

Asymptotics of Discrete Painlevé V transcendents via the Riemann–Hilbert Approach

We study a system of discrete Painlevé V equations via the Riemann–Hilbert approach. We begin with an isomonodromy problem for dPV, which admits a discrete Riemann–Hilbert problem formulation. The asymptotics of the discrete Riemann–Hilbert problem is derived via the nonlinear steepest descent method of Deift and Zhou. In the analysis, a parametrix is constructed in terms of specific Painlevé V transcendents. As a result, the asymptotics of the dPV transcendents are represented in terms of the PV transcendents. In the special case, our result confirms a conjecture of Borodin, that the difference Schlesinger equations converge to the differential Schlesinger equations at the solution level.     

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in an L-shaped domain

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in an L-shaped domain is considered for when the solution has singularities at the corners of the domain. The densification of the Shishkin mesh near the inner corner where different boundary conditions meet is such that the solution obtained by the classical five-point difference scheme converges to the solution of the initial problem in the mesh norm L _?? ^h uniformly with respect to the small parameter with almost second order, i.e., as a smooth solution. Numerical analysis confirms the theoretical result.