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

Abstract

A new technique is proposed for the solution of the Riemann–Hilbert problem with the Chebotarev–Khrapkov matrix coefficient (Formula presented.) zero-trace polynomial matrix. This problem has numerous applications in elasticity and diffraction theory. The main feature of the method is the removal of 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 the finding of the (Formula presented.) zeros of the Baker–Akhiezer function (Formula presented.) 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-(Formula presented.) equations.