Inverse scattering transforms for non-local reverse-space matrix non-linear Schrödinger equations

The aim of the paper is to explore non-local reverse-space matrix non-linear Schrödinger equations and their inverse scattering transforms. Riemann–Hilbert problems are formulated to analyse the inverse scattering problems, and the Sokhotski–Plemelj formula is used to determine Gelfand–Levitan–Marchenko-type integral equations for generalised matrix Jost solutions. Soliton solutions are constructed through the reflectionless transforms associated with poles of the Riemann–Hilbert problems.

Inverse scattering for nonlocal reverse-space multicomponent nonlinear Schrödinger equations

The paper aims to discuss nonlocal reverse-space multicomponent nonlinear Schrödinger equations and their inverse scattering transforms. The inverse scattering problems are analyzed by means of Riemann–Hilbert problems, and Gelfand–Levitan–Marchenko-type integral equations for generalized matrix Jost solutions are determined by the Sokhotski–Plemelj formula. Soliton solutions are generated from the reflectionless transforms associated with zeros of the Riemann–Hilbert problems.

Mixed Boundary Value Problem for an Anisotropic Thermoelastic Half-Space Containing Thin Inhomogeneities

The paper presents a rigorous and straightforward approach for obtaining the 2D boundary integral equations for a thermoelastic half-space containing holes, cracks and thin foreign inclusions. It starts from the Cauchy integral formula and the extended Stroh formalism which allows writing the general solution of thermoelastic problems in terms of certain analytic functions. In addition, with the help of it, it is possible to convert the volume integrals included in the equation into contour integrals, which, in turn, will allow the use of the method of boundary elements. For modelling of solids with thin inhomogeneities, a coupling principle for continua of different dimensions is used. Applying the theory of complex variable functions, in particular, Cauchy integral formula and Sokhotski–Plemelj formula, the Somigliana type boundary integral equations are constructed for thermoelastic anisotropic half-space. The obtained integral equations are introduced into the modified boundary element method. A numerical analysis of the influence of boundary conditions on the half-space boundary and relative rigidity of the thin inhomogeneity on the intensity of stresses at the inclusions is carried out.

Linear BVPs and SIEs for Generalized Regular Functions in Clifford Analysis

We study some properties of a regular function in Clifford analysis and generalize Liouville theorem and Plemelj formula with values in Clifford algebra An(R). By means of the classical Riemann boundary value problem and of the theory of a regular function, we discuss some boundary value problems and singular integral equations in Clifford analysis and obtain the explicit solutions and the conditions of solvability. Thus, the results in this paper will be of great significance for the study of improving and developing complex analysis, integral equation, and boundary value theory.

Generalized Cauchy integrals on the plane

The integrals with homogeneous-difference kernels are considered on a smooth contour. The boundary properties of the integrals are described in the H?lder space. An analogue of the known Sokhotski-Plemelj formula is obtained. Moreover, the differentiation formula of these integrals is also given.

A Boundary Value Problem for Bihypermonogenic Functions in Clifford Analysis

This paper deals with a nonlinear boundary value problem for bihypermonogenic functions in Clifford analysis. The integrals of quasi-Cauchy’s type and Plemelj formula for bihypermonogenic functions are firstly reviewed briefly. The nonlinear Riemmann boundary value problem for bihypermonogenic functions is discussed and the existence of solutions is obtained, which also indicates that the linear boundary value problem has a unique solution.

Noether Theorem of a Kind of Singular Integral Equation with Hilbert Kernel on Open Arcs

We discussed a kind of singular integral equation with Hilbert kernel on open arcs lying in a period strip. By using the method of complex functions, we obtained the extended Plemelj Formula with Hilbert kernel, and based on this, we obtained the general solutions and the solvable conditions for this kind of characteristic singular integral equation with Hilbert kernel on open arcs.