Vector Riemann-Hilbert Problem Research Articles

Scattering by a Perforated Sandwich Panel: Method of Riemann Surfaces

Summary The model problem of scattering of a sound wave by an infinite plane structure formed by a semi-infinite acoustically hard screen and a semi-infinite sandwich panel perforated from one side and covered by a membrane from the other is exactly solved. The model is governed by two Helmholtz equations for the velocity potentials in the upper and lower half-planes coupled by the Leppington effective boundary condition and the equation of vibration of a membrane in a fluid. Two methods of solution are proposed and discussed. Both methods reduce the problem to an order-2 vector Riemann–Hilbert problem. The matrix coefficients have different entries, have the Chebotarev–Khrapkov structure and share the same order-4 characteristic polynomial. Exact Wiener–Hopf matrix factorization requires solving a scalar Riemann–Hilbert on an elliptic surface and the associated genus-1 Jacobi inversion problem solved in terms of the associated Riemann θ-function. Numerical results for the absolute value of the total velocity potentials are reported and discussed.

A free boundary problem of elasticity in an angle and a vector Riemann–Hilbert problem on a torus

An inverse problem of elasticity of reconstructing the shape of an inclusion placed in a wedge, when the whole inclusion is subjected to a prescribed antiplane uniform stress, is considered. The doubly connected physical domain is conformally mapped onto a parametric plane cut along two finite segments on the real axis. Determination of such a map requires solving a vector Riemann–Hilbert problem of the theory of holomorphic functions on a symmetric genus-1 Riemann surface. The main steps of the method include factorization of a discontinuous function on a torus, solution of the associated Jacobi inversion problem, representation of the unknown vector-functions in terms of Weierstrass integrals, and reduction of the vector Riemann–Hilbert problem to a single integral kernel of the second kind with a discontinuous kernel on the elliptic surface. An integral representation of the conformal map in terms of the solution to the integral equation is given. In addition to four dimensionless problem parameters, the conformal map possesses five free parameters. Numerical results, which reconstruct the inclusion shape for sample sets of the parameters, are reported and discussed.

The binary Darboux transformation revisited and KdV solitons on arbitrary short‐range backgrounds

AbstractIn the KdV context, we revisit the classical Darboux transformation in the framework of the vector Riemann–Hilbert problem. This readily yields a version of the binary Darboux transformation providing a short‐cut to explicit formulas for solitons traveling on a wide range of background solutions. Our approach also links the binary Darboux transformation and the double commutation method.

Riemann–Hilbert problem on a hyperelliptic surface and uniformly stressed inclusions embedded into a half-plane subjected to antiplane strain

An inverse problem of the elasticity of n elastic inclusions embedded into an elastic half-plane is analysed. The boundary of the half-plane is free of traction. The half-plane and the inclusions are subjected to antiplane shear, and the conditions of ideal contact hold in the interfaces between the inclusions and the half-plane. The shapes of the inclusions are not prescribed and have to be determined by enforcing uniform stresses inside the inclusions. The method of conformal mappings from a slit domain onto the ( n + 1 ) -connected physical domain is worked out. It is shown that to recover the map and the shapes of the inclusions, one needs to solve a vector Riemann–Hilbert problem on a genus- n hyperelliptic surface. In a particular case of loading, the vector problem reduces to two scalar Riemann–Hilbert problems on n + 1 slits on a hyperelliptic surface. In the elliptic case, in addition to three parameters of the model, the conformal map possesses a free geometric parameter. The results of numerical tests in the elliptic case show the impact of these parameters on the inclusion shape.

Debonded arc-shaped interface conducting rigid line inclusions in piezoelectric composites

We consider an arc-shaped conducting rigid line inclusion located at the interface between a circular piezoelectric inhomogeneity and an unbounded piezoelectric matrix subjected to remote uniform anti-plane shear stresses and in-plane electric fields. Moreover, one side of the rigid line inclusion has become fully debonded from the matrix or the inhomogeneity leading to the formation of an insulating crack. After the introduction of two sectionally holomorphic vector functions, the problem is reduced to a vector Riemann–Hilbert problem, which can be decoupled sequentially by repeated application of the orthogonality relations between the eigenvectors for two corresponding generalized eigenvalue problems.

OUP accepted manuscript

The interaction of a thin rigid inclusion with a finite crack is studied. Two plane problems of elasticity are considered. The first one concerns the case when the upper side of the inclusion is completely debonded from the matrix, and the crack penetrates into the medium. In the second model, the upper side of the inclusion is partly separated from the matrix, that is the crack length $2a$ is less than $2b$, the inclusion length. It is shown that both problems are governed by a singular integral equation of the same structure. Derivation of the closed-form solution of this integral equation is the main result of the paper. The solution is found by solving the associated vector Riemann-Hilbert problem with the Chebotarev-Khrapkov matrix coefficient. A feature of the method proposed is that the vector Riemann-Hilbert problem is set on a finite segment, while the original Khrapkov method of matrix factorization is developed for a closed contour. In the case, when the crack and inclusion lengths are the same, the solution is derived by passing to the limit $b/a\to 1$. It is demonstrated that the limiting case $a=b$ is unstable, and when $a<b$, and the crack tips approach the inclusion ends, the crack tends to accelerate in order to penetrate into the matrix.

The Initial-boundary Value Problem for the Ostrovsky-Vakhnenko Equation on the Half-line

We analyze an initial-boundary value problem for the Ostrovsky-Vakhnenko equation on the half-line. This equation can be viewed as the short wave model for the Degasperis-Procesi (DP) equation. We show that the solution u(x,t) can be recovered from its initial and boundary values via the solution of a vector Riemann-Hilbert problem formulated in the plane of a complex spectral parameter z.

Singular integral equations with two fixed singularities and applications to fractured composites

A symmetric characteristic singular integral equation with two fixed singularities at the endpoints in the class of functions bounded at the ends is analyzed. It reduces to a vector Hilbert problem for a half-disc and then to a vector Riemann-Hilbert problem on a real axis with a piecewise constant matrix coefficient that has two points of discontinuity. A condition of solvability and a closed-form solution to the integral equation are derived. For the Chebyshev polynomials of the first kind in the right hand-side, the solution of the integral equation is expressed in terms of two nonorthogonal polynomials with associated weights. Based on this new generalized spectral relation for the singular operator with two fixed singularities an approximate solution to the complete singular integral equation is derived by recasting it as an infinite system of linear algebraic equations of the second kind. The method is illustrated by solving two problems of fracture mechanics, the antiplane and plane strain problems for a finite crack in a composite plane. The plane is formed by a strip and two half-planes; the elastic constants of the strip are different from those of the half-planes. The crack is orthogonal to the interfaces, and it is located in the strip with the ends lying in the interfaces. Numerical results are reported and discussed.

Vector Riemann–Hilbert problem with almost periodic and meromorphic coefficients and applications

The vector Riemann–Hilbert problem is analysed when the entries of its matrix coefficient are meromorphic and almost periodic functions. Three cases for the meromorphic functions, when they have (i) a finite number of poles and zeros (rational functions), (ii) periodic poles and zeros, and (iii) an infinite number of non-periodic zeros and poles, are considered. The first case is illustrated by the heat equation for a composite rod with a finite number of discontinuities and a system of convolution equations; both problems are solved explicitly. In the second case, a Wiener–Hopf factorization is found in terms of the hypergeometric functions, and the exact solution of a mixed boundary value problem for the Laplace equation in a wedge is derived. In the last case, the Riemann–Hilbert problem reduces to an infinite system of linear algebraic equations with the exponential rate of convergence. As an example, the Neumann boundary value problem for the Helmholtz equation in a strip with a slit is analysed.

The Baker–Akhiezer Function and Factorization of the Chebotarev–Khrapkov Matrix

A new technique is proposed for the solution of the Riemann-Hilbert problem with the Chebotarev-Khrapkov matrix coefficient $G(t)=\alpha_1(t)I+\alpha_2(t)Q(t)$, $\alpha_1(t), \alpha_2(t)\in H(L)$, $Q(t)$ is a $2\times 2$ zero-trace polynomial matrix, and $I$ is the unit matrix. This problem has numerous applications in elasticity and diffraction theory. The main feature of the method is the removal of the essential singularities of the solution to the associated homogeneous scalar Riemann-Hilbert problem on the hyperelliptic surface of an algebraic function by means of the Baker-Akhiezer function. The consequent application of this function for the derivation of the general solution to the vector Riemann-Hilbert problem requires finding of the $\rho$ zeros of the Baker-Akhiezer function ($\rho$ is the genus of the surface). These zeros are recovered through the solution to the associated Jacobi problem of inversion of abelian integrals or, equivalently, the determination of the zeros of the associated degree-$\rho$ polynomial and solution of a certain linear algebraic system of $\rho$ equations.

Diffraction of an obliquely incident electromagnetic wave by an impedance right-angled concave wedge

Scattering of a plane electromagnetic wave by an anisotropic impedance right-angled concave wedge at skew incidence is analysed. A closed-form solution is derived by reducing the problem to a symmetric order-2 vector Riemann–Hilbert problem (RHP) on the real axis. The problem of matrix factorization leads to a scalar RHP on a genus-3 Riemann surface. Its solution is derived by the Weierstrass integrals. Due to a special symmetry of the problem the associated Jacobi inversion problem is solved in terms of elliptic integrals, not a genus-3 Riemann θ-function. The electric and magnetic field components are expressed through the Sommerfeld integrals, and the incident and reflected waves are recovered.

Subsonic propagation of a crack parallel to the boundary of a half-plane

A two-dimensional steady-state model problem on a half-plane with a semi-infinite crack propagating at constant speed parallel to the boundary of the half-plane is considered. The crack faces are subjected to normal and tangential loads, while the boundary of the half-plane is free of traction. The problem is formulated as an order-2 vector Riemann–Hilbert problem and then reduced to a system of singular integral equations in a semi-infinite segment with respect to the derivatives of the displacement jumps. The solution to the system of integral equations is represented in a series form in terms of an orthonormal basis of the associated Hilbert space, the orthonormal Jacobi polynomials. The coefficients of the expansions solve an infinite system of linear algebraic equations of the second kind. The stress intensity factors and the weight functions are determined and computed. The Griffith energy criterion is applied to derive a crack growth criterion in terms of the K I - and K II -stress intensity factors, the crack speed, the shear and dilatational waves speeds and the Griffith material constant.

Long-time asymptotics of the periodic Toda lattice under short-range perturbations

We compute the long-time asymptotics of periodic (and slightly more generally of algebro-geometric finite-gap) solutions of the doubly infinite Toda lattice under a short-range perturbation. In particular, we prove that the perturbed lattice asymptotically approaches a modulated lattice. More precisely, let g be the genus of the hyperelliptic curve associated with the unperturbed solution. We show that, apart from the phenomenon of solitons travelling in a quasi-periodic background, the n/t-pane contains g + 2 areas where the perturbed solution is close to a finite-gap solution on the same isospectral torus. In between there are g + 1 regions where the perturbed solution is asymptotically close to a modulated lattice which undergoes a continuous phase transition (in the Jacobian variety) and which interpolates between these isospectral solutions. In the special case of the free lattice (g = 0), the isospectral torus consists of just one point and we recover the known result. Both the solutions in the isospectral torus and the phase transition are explicitly characterized in terms of Abelian integrals on the underlying hyperelliptic curve. Our method relies on the equivalence of the inverse spectral problem to a vector Riemann–Hilbert problem defined on the hyperelliptic curve and generalizes the so-called nonlinear stationary phase/steepest descent method for Riemann–Hilbert problem deformations to Riemann surfaces.

Integro-Differential Equation for a Finite Crack in a Strip with Surface Effects

We present a mathematical investigation of a nonstandard boundary value problem for Laplace's equation in an infinite strip containing a finite crack. This unusual problem arises when surface elasticity is incorporated in the description of deformation of the crack faces. We use the Fourier transform to reduce the original boundary value problem to an integral equation that is further shown to reduce to a vector Riemann―Hilbert problem. The latter is solved analytically. The solution consists of two parts. The first is found explicitly, and the second is in a series form. The series converge exponentially, and the series coefficients are obtained by solving an infinite system. It is shown that the rate of convergence of an approximate solution to the exact one is exponential. We illustrate our solution with a particular example which, among other interesting phenomena, reveals the effect of the surface mechanics on the shape of the crack face and shows that the stress is bounded at the crack tip.

Riemann–Hilbert Approach to the Elastodynamic Equation: Part I

We develop the Riemann–Hilbert (RH) approach to scattering problems in elastic media. The approach is based on the RH method introduced in the 1990s by Fokas (A unified approach to boundary value problems, CBMS-SIAM, 2008) for studying boundary problems for linear and integrable nonlinear PDEs. A suitable Lax pair formulation of the elastodynamic equation is obtained. The integral representations derived from this Lax pair are applied to Rayleigh wave propagation in an elastic half space and quarter space. The latter problem is reduced to the analysis of a certain underdetermined RH problem. We show that the problem can be re-formulated as a well-determined vector Riemann–Hilbert problem with a shift posed on a torus.

Subsonic semi-infinite crack with a finite friction zone in a bimaterial

Propagation of a semi-infinite crack along the interface between an elastic half-plane and a rigid half-plane is analyzed. The crack advances at constant subsonic speed. It is assumed that, ahead of the crack, there is a finite segment where the conditions of Coulomb friction law are satisfied. The contact zone of unknown a priori length propagates with the same speed as the crack. The problem reduces to a vector Riemann–Hilbert problem with a piece-wise constant matrix coefficient discontinuous at three points, 0, 1, and ∞ . The problem is solved exactly in terms of Kummer's solutions of the associated hypergeometric differential equation. Numerical results are reported for the length of the contact friction zone, the stress singularity factor, the normal displacement u 2 , and the dynamic energy release rate G. It is found that in the case of frictionless contact for both the sub-Rayleigh and super-Rayleigh regimes, G is positive and the stress intensity factor K II does not vanish. In the sub-Rayleigh case, the normal displacement is positive everywhere in the opening zone. In the super-Rayleigh regime, there is a small neighborhood of the ending point of the open zone where the normal displacement is negative.

The use of case's method to solve the linearized BGK equations for the temperature-jump problem

The temperature-jump problem in a rarefied gas occupying a half-space with given temperature gradient at infinity is considered. Case's method is used to find the eigenvectors and eigenvalues of the appropriate transport operators. A completeness theorem for the family of eigenvectors is proved by solving the vector Riemann-Hilbert problem with matrix coefficients. The matrix that puts the boundary-value problem coefficient into diagonal form is analytic in the complex plane with two cuts joining two pairs of branch pints. This necessitates solving an additional boundary-value problem on the cuts, by means of which a fundamental matrix is constructed. A solution of the problem is constructed using this matrix, and, as an application, an exact formula is obtained for calculating the temperature jump by quadratures.