Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 054, 11 pages      math-ph/0703072      https://doi.org/10.3842/SIGMA.2007.054
Contribution to the Vadim Kuznetsov Memorial Issue

Bäcklund-Darboux Transformation for Non-Isospectral Canonical System and Riemann-Hilbert Problem

Alexander Sakhnovich
Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria

Received October 25, 2006, in final form March 19, 2007; Published online March 25, 2007

Abstract
A GBDT version of the Bäcklund-Darboux transformation is constructed for a non-isospectral canonical system, which plays essential role in the theory of random matrix models. The corresponding Riemann-Hilbert problem is treated and some explicit formulas are obtained. A related inverse problem is formulated and solved.

Key words: Bäcklund-Darboux transformation; canonical system; random matrix theory.

pdf (229 kb)   ps (148 kb)   tex (12 kb)

References

  1. Cieslinski J., An effective method to compute N-fold Darboux matrix and N-soliton surfaces, J. Math. Phys. 32 (1991), 2395-2399.
  2. Deift P., Applications of a commutation formula, Duke Math. J. 45 (1978), 267-310.
  3. Deift P., Its A., Zhou X., A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics, Ann. of Math. (2) 146 (1997), 149-235.
  4. Deift P.A., Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, in Courant Lecture Notes in Mathematics, Vol. 3, AMS, Providence, RI, 1999.
  5. Fritzsche B., Kirstein B., Sakhnovich A.L., Completion problems and scattering problems for Dirac type differential equations with singularities, J. Math. Anal. Appl. 317 (2006), 510-525, math.SP/0409424.
  6. Gesztesy F., A complete spectral characterization of the double commutation method, J. Funct. Anal. 117 (1993), 401-446.
  7. Gohberg I., Kaashoek M.A., Sakhnovich A.L., Scattering problems for a canonical system with a pseudo-exponential potential, Asymptotic Analysis 29 (2002), 1-38.
  8. Gu C., Hu H., Zhou Z., Darboux transformations in integrable systems, Math. Phys. Stud., Vol. 26, Springer, Dordrecht, 2005.
  9. Kuznetsov V.B., Petrera M., Ragnisco O., Separation of variables and Bäcklund transformations for the symmetric Lagrange top, J. Phys. A: Math. Gen. 37 (2004), 8495-8512, nlin.SI/0403028.
  10. Kuznetsov V.B., Salerno M., Sklyanin E.K., Quantum Bäcklund transformation for the integrable DST model, J. Phys. A: Math. Gen. 33 (2000), 171-189, solv-int/9908002.
  11. Marchenko V.A., Nonlinear equations and operator algebras, Reidel Publishing Co., Dordrecht, 1988.
  12. Matveev V.B., Salle M.A., Darboux transformations and solitons, Springer, Berlin, 1991.
  13. Miura R. (Editor), Bäcklund transformations, Lecture Notes in Math., Vol. 515, Springer, Berlin, 1976.
  14. Sakhnovich A.L., Iterated Bäcklund-Darboux transformation and transfer matrix-function (nonisospectral case), Chaos Solitons Fractals 7 (1996), 1251-1259.
  15. Sakhnovich A.L., Iterated Bäcklund-Darboux transform for canonical systems, J. Funct. Anal. 144 (1997), 359-370.
  16. Sakhnovich L.A., Operators, similar to unitary operators, with absolutely continuous spectrum, Funct. Anal. Appl. 2 (1968), 48-60.
  17. Sakhnovich L.A., On the factorization of the transfer matrix function, Sov. Math. Dokl. 17 (1976), 203-207.
  18. Sakhnovich L.A., Factorisation problems and operator identities, Russian Math. Surv. 41 (1986), 1-64.
  19. Sakhnovich L.A., Spectral theory of canonical differential systems. Method of operator identities, Oper. Theory Adv. Appl., Vol. 107, Birkhäuser, Basel - Boston, 1999.
  20. Sakhnovich L.A., Integrable operators and canonical differential systems, Math. Nachr. 280 (2007), 205-220, math.FA/0403490.
  21. Teschl G., Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, Vol. 72, AMS, Providence, RI, 2000.
  22. Wiener N., Extrapolation, interpolation, and smoothing of stationary time series, Chapman and Hall Ltd., London, 1949.
  23. Zakharov V.E., Mikhailov A.V., On the integrability of classical spinor models in two-dimensional space-time, Comm. Math. Phys. 74 (1980), 21-40.

Previous article   Next article   Contents of Volume 3 (2007)