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 $G(t)=α_1(t)I+α_2(t)Q(t)$, $α_1(t), α_2(t)\in H(L)$, $Q(t)$ 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 $ρ$ zeros of the Baker-Akhiezer function ($ρ$ 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-$ρ$ polynomial and solution of a certain linear algebraic system of $ρ$ equations.