Letters in Mathematical Physics

, Volume 104, Issue 11, pp 1365–1384 | Cite as

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 \({G(t) = \alpha_{1}(t)I + \alpha_{2}(t)Q(t)}\), \({\alpha_{1}(t), \alpha_{2}(t) \in H(L)}\), I = diag{1, 1}, Q(t) is a \({2\times2}\) 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 \({\rho}\) 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.

Mathematics Subject Classification

30E25 30F99 45E 


Riemann–Hilbert problem Baker–Akhiezer function Riemann surfaces 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Antipov Y.A.: An exact solution of the 3-D-problem on an interface semi-infinite plane crack. J. Mech. Phys. Solids 47, 1051–1093 (1999)MathSciNetCrossRefMATHADSGoogle Scholar
  2. 2.
    Antipov Y.A.: Solution by quadratures of the problem of a cylindrical crack by the method of matrix factorization. IMA J. Appl. Math. 66, 591–619 (2001)MathSciNetCrossRefMATHADSGoogle Scholar
  3. 3.
    Antipov Y.A.: A symmetric Riemann–Hilbert problem for order-4 vectors in diffraction theory. Q. J. Mech. Appl. Math. 63, 349–374 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Antipov Y.A.: A genus-3 Riemann–Hilbert problem and diffraction of a wave by orthogonal resistive half-planes. Comput. Methods Funct. Theory 11, 439–462 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Antipov Y.A., Moiseev N.G.: Exact solution of the plane problem for a composite plane with a cut across the boundary between two media. J. Appl. Math. Mech. 55, 531–539 (1991)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Antipov Y.A., Silvestrov V.V.: Factorization on a Riemann surface in scattering theory. Q. J. Mech. Appl. Math. 55, 607–654 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Antipov Y.A., Silvestrov V.V.: Vector functional-difference equation in electromagnetic scattering. IMA J. Appl. Math. 69, 27–69 (2004)MathSciNetCrossRefMATHADSGoogle Scholar
  8. 8.
    Antipov Y.A., Silvestrov V.V.: Electromagnetic scattering from an anisotropic impedance half plane at oblique incidence: the exact solution. Q. J. Mech. Appl. Math. 59, 211–251 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Büyükaksoy A., Serbest A.H: Matrix Wiener–Hopf methods applications to some diffraction problems. In: Hashimoto, M., Ideman, M., Tretyakov, O.A. (eds.) Analytical and Numerical Methods in Electromagnetic Wave Theory, pp. 257–315. Science House Co. Ltd, Tokyo (1993)Google Scholar
  10. 10.
    Chebotarev G.N.: On closed-form solution of a Riemann boundary value problem for n pairs of functions. Uchen. Zap. Kazan. Univ. 116, 31–58 (1956)Google Scholar
  11. 11.
    Chebotarev N.G.: Theory of Algebraic Functions. OGIZ, Moscow (1948)Google Scholar
  12. 12.
    Daniele V.G.: On the solution of vector Wiener–Hopf equations occurring in scattering problems. Radio Sci. 19, 1173–1178 (1984)CrossRefADSGoogle Scholar
  13. 13.
    Dubrovin B.A.: The inverse scattering problem for periodic finite-zone potentials. Funct. Anal. Appl. 9, 61–62 (1975)CrossRefMATHGoogle Scholar
  14. 14.
    Dubrovin B.A.: Theta functions and non-linear equations. Russ. Math. Surv. 36(2), 11–92 (1981)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Dubrovin B.A., Matveev V.B., Novikov S.P.: Nonlinear equations of Korteweg–de Vries type, finite-band linear operators and Abelian varieties. Russ. Math. Surv. 31(1), 59–146 (1976)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hurd R.A., Lüneburg E.: Diffraction by an anisotropic impedance half plane. Can. J. Phys. 63, 1135–1140 (1985)CrossRefADSGoogle Scholar
  17. 17.
    Its A.R., Matveev V.B.: Schrödinger operators with the finite-band spectrum and the N-soliton solutions of the Korteweg–de Vries equation. Theor. Math. Phys. 23, 343–355 (1975)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Jones C.M.A.: Scattering by a semi-infinite sandwich panel perforated on one side. Proc. R. Soc. A 454, 465–479 (1990)CrossRefADSGoogle Scholar
  19. 19.
    Khrapkov A.A.: Certain cases of the elastic equilibrium of an infinite wedge with a nonsymmetric notch at the vertex, subjected to concentrated forces. J. Appl. Math. Mech. 35, 625–637 (1971)CrossRefMATHGoogle Scholar
  20. 20.
    Krichever I.M.: Methods of algebraic geometry in the theory of nonlinear equations. Russ. Math. Surv. 32(6), 185–213 (1971)CrossRefGoogle Scholar
  21. 21.
    Lüneburg E., Serbest A.H.: Diffraction of an obliquely incident plane wave by a two-face impedance half plane: Wiener–Hopf approach. Radio Sci. 35, 1361–1374 (2000)CrossRefADSGoogle Scholar
  22. 22.
    Matveev V.B.: 30 years of finite-gap integration theory. Phil. Trans. R. Soc. A 366, 837–875 (2008)CrossRefMATHADSGoogle Scholar
  23. 23.
    Moiseev N.G.: Factorization of matrix functions of special form. Sov. Math. Dokl. 39, 264–267 (1989)MathSciNetMATHGoogle Scholar
  24. 24.
    Moiseyev N.G., Popov G.Y.: Exact solution of the problem of bending of a semi-infinite plate completely bonded to an elastic half-space. Izv. Akad. Nauk SSSR, Solid Mech. 25, 113–125 (1990)Google Scholar
  25. 25.
    Rawlins A.D.: Two waveguide trifurcation problems. Math. Proc. Camb. Philos. Soc. 121, 555–573 (1995)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Springer G.: Introduction to Riemann Surfaces. Addison-Wesley, Reading (1956)Google Scholar
  27. 27.
    Vekua N.P.: Systems of Singular Integral Equations. Noordhoff, Groningen (1967)MATHGoogle Scholar
  28. 28.
    Zverovich E.I.: Boundary value problems in the theory of analytic functions in Hölder classes on Riemann surfaces. Russ. Math. Surv. 26(1), 117–192 (1971)CrossRefGoogle Scholar
  29. 29.
    Zverovich E.I.: The problem of linear conjugation on a closed Riemann surface. Complex Anal. Oper. Theory 2, 709–732 (2008)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA

Personalised recommendations