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

SIGMA 12 (2016), 054, 30 pages      arXiv:1511.08098      https://doi.org/10.3842/SIGMA.2016.054
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

Multidimensional Toda Lattices: Continuous and Discrete Time

Alexander I. Aptekarev a, Maxim Derevyagin b, Hiroshi Miki c and Walter Van Assche d
a) Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, 125047 Moscow, Russia
b) University of Mississippi, Department of Mathematics, Hume Hall 305, P. O. Box 1848, University, MS 38677-1848, USA
c) Doshisha University, Department of Electronics, Faculty of Science and Engineering, Kyotanabe city, Kyoto 610 0394, Japan
d) KU Leuven, Department of Mathematics, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium

Received January 05, 2016, in final form June 01, 2016; Published online June 13, 2016

Abstract
In this paper we present multidimensional analogues of both the continuous- and discrete-time Toda lattices. The integrable systems that we consider here have two or more space coordinates. To construct the systems, we generalize the orthogonal polynomial approach for the continuous and discrete Toda lattices to the case of multiple orthogonal polynomials.

Key words: multiple orthogonal polynomials; orthogonal polynomials; recurrence relations; Toda equation; discrete integrable system; Toda lattice.

pdf (509 kb)   tex (36 kb)

References

  1. Adler M., Horozov E., van Moerbeke P., The Pfaff lattice and skew-orthogonal polynomials, Int. Math. Res. Not. 1999 (1999), 569-588, solv-int/9903005.
  2. Adler M., van Moerbeke P., Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems, Comm. Math. Phys. 207 (1999), 589-620, nlin.SI/0009002.
  3. Adler V.E., Discrete equations on planar graphs, J. Phys. A: Math. Gen. 34 (2001), 10453-10460.
  4. Álvarez-Fernández C., Fidalgo Prieto U., Mañas M., Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy, Adv. Math. 227 (2011), 1451-1525, arXiv:1004.3916.
  5. Angelesco A., Sur deux extensions des fractions continues algébriques, C. R. Acad. Sci. Paris 168 (1919), 262-265.
  6. Aptekarev A.I., Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 423-447.
  7. Aptekarev A.I., Spectral problems of high-order recurrences, in Spectral theory and differential equations, Amer. Math. Soc. Transl. Ser. 2, Vol. 233, Amer. Math. Soc., Providence, RI, 2014, 43-61.
  8. Aptekarev A.I., The Mhaskar-Saff variational principle and location of the shocks of certain hyperbolic equations, in Modern Trends in Constructive Function Theory, Contemporary Mathematics, Vol. 661, Amer. Math. Soc., Providence, RI, 2016, 167-186.
  9. Aptekarev A.I., Bleher P.M., Kuijlaars A.B.J., Large $n$ limit of Gaussian random matrices with external source. II, Comm. Math. Phys. 259 (2005), 367-389, math-ph/0408041.
  10. Aptekarev A.I., Branquinho A., Padé approximants and complex high order Toda lattices, J. Comput. Appl. Math. 155 (2003), 231-237.
  11. Aptekarev A.I., Branquinho A., Marcellán F., Toda-type differential equations for the recurrence coefficients of orthogonal polynomials and Freud transformation, J. Comput. Appl. Math. 78 (1997), 139-160.
  12. Aptekarev A.I., Branquinho A., Van Assche W., Multiple orthogonal polynomials for classical weights, Trans. Amer. Math. Soc. 355 (2003), 3887-3914.
  13. Aptekarev A.I., Derevyagin M., Van Assche W., On 2D discrete Schrödinger operators associated with multiple orthogonal polynomials, J. Phys. A: Math. Theor. 48 (2015), 065201, 16 pages, arXiv:1410.1332.
  14. Aptekarev A.I., Derevyagin M., Van Assche W., Discrete integrable systems generated by Hermite-Padé approximants, Nonlinearity 29 (2016), 1487-1506, arXiv:1409.4053.
  15. Barrios Rolanía D., Branquinho A., Foulquié Moreno A., On the relation between the full Kostant-Toda lattice and multiple orthogonal polynomials, J. Math. Anal. Appl. 377 (2011), 228-238, arXiv:0911.2856.
  16. Ben Cheikh Y., Douak K., On the classical $d$-orthogonal polynomials defined by certain generating functions. I, Bull. Belg. Math. Soc. Simon Stevin 7 (2000), 107-124.
  17. Ben Cheikh Y., Douak K., On the classical $d$-orthogonal polynomials defined by certain generating functions. II, Bull. Belg. Math. Soc. Simon Stevin 8 (2001), 591-605.
  18. Bernstein L., The Jacobi-Perron algorithm - Its theory and application, Lecture Notes in Math., Vol. 207, Springer-Verlag, Berlin - New York, 1971.
  19. Bleher P.M., Kuijlaars A.B.J., Random matrices with external source and multiple orthogonal polynomials, Int. Math. Res. Not. 2004 (2004), 109-129, math-ph/0307055.
  20. Bobenko A.I., Suris Yu.B., Integrable systems on quad-graphs, Int. Math. Res. Not. 2002 (2002), 573-611, nlin.SI/0110004.
  21. Bueno M.I., Marcellán F., Darboux transformation and perturbation of linear functionals, Linear Algebra Appl. 384 (2004), 215-242.
  22. Coussement J., Kuijlaars A.B.J., Van Assche W., Direct and inverse spectral transform for the relativistic Toda lattice and the connection with Laurent orthogonal polynomials, Inverse Problems 18 (2002), 923-942, math.CA/0204155.
  23. Coussement J., Van Assche W., An extension of the Toda lattice: a direct and inverse spectral transform connected with orthogonal rational functions, Inverse Problems 20 (2004), 297-318.
  24. Coussement J., Van Assche W., A continuum limit of the relativistic Toda lattice: asymptotic theory of discrete Laurent orthogonal polynomials with varying recurrence coefficients, J. Phys. A: Math. Gen. 38 (2005), 3337-3366.
  25. Douak K., Maroni P., Une caractérisation des polynômes $d$-orthogonaux ''classiques'', J. Approx. Theory 82 (1995), 177-204.
  26. Fidalgo Prieto U., López Lagomasino G., Nikishin systems are perfect, Constr. Approx. 34 (2011), 297-356, arXiv:1001.0554.
  27. Filipuk G., Haneczok M., Van Assche W., Computing recurrence coefficients of multiple orthogonal polynomials, Numer. Algorithms 70 (2015), 519-543, arXiv:1406.0364.
  28. Gragg W.B., The Padé table and its relation to certain algorithms of numerical analysis, SIAM Rev. 14 (1972), 1-16.
  29. Grammaticos B., Kosmann-Schwarzbach Y., Tamizhmani T. (Editors), Discrete integrable systems, Lecture Notes in Phys., Vol. 644, Springer-Verlag, Berlin, 2004.
  30. Hermite C., Sur la fonction exponentielle, C. R. Acad. Sci. Paris 77 (1873), 18-24; 74-79; 226-233.
  31. Hirota R., Conserved quantities of ''random-time Toda equation'', J. Phys. Soc. Japan 66 (1997), 283-284.
  32. Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2005.
  33. Jack I., Jones D.R.T., Panvel J., Quantum non-abelian Toda field theories, Internat. J. Modern Phys. A 9 (1994), 3631-3656, hep-th/9308080.
  34. Jacobi C.G.J., Heine E., Ueber die Auslösung der Gleichung $a_l x_1 + a_2x_2 + \cdots + a_nx_n = f\cdot u$, J. Reine Angew. Math. 69 (1868), 1-28.
  35. Jacobi C.G.J., Heine E., Allgemeine Theorie der kettenbruchähnlichen Algorithmen, in welchen jede Zahl aus drei vorhergehenden gebildet wird, J. Reine Angew. Math. 69 (1868), 29-64.
  36. Kac M., van Moerbeke P., On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices, Adv. Math. 16 (1975), 160-169.
  37. Mahler K., Perfect systems, Compositio Math. 19 (1968), 95-166.
  38. Manakov S.V., Santini P.M., Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking, J. Phys. A: Math. Theor. 44 (2011), 345203, 19 pages, arXiv:1011.2619.
  39. Miki H., Tsujimoto S., Vinet L., Zhedanov A., An algebraic model for the multiple Meixner polynomials of the first kind, J. Phys. A: Math. Theor. 45 (2012), 325205, 11 pages, arXiv:1203.0357.
  40. Miki H., Vinet L., Zhedanov A., Non-Hermitian oscillator Hamiltonians and multiple Charlier polynomials, Phys. Lett. A 376 (2011), 65-69, arXiv:1106.5243.
  41. Moser J., Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16 (1975), 197-220.
  42. Ndayiragije F., Van Assche W., Multiple Meixner polynomials and non-Hermitian oscillator Hamiltonians, J. Phys. A: Math. Gen. 46 (2013), 505201, 17 pages, arXiv:1310.0982.
  43. Nikishin E.M., Simultaneous Padé approximants, Math. USSR Sb. 41 (1982), 409-425.
  44. Nikishin E.M., Sorokin V.N., Rational approximations and orthogonality, Translations of Mathematical Monographs, Vol. 92, Amer. Math. Soc., Providence, RI, 1991.
  45. Nuttall J., Asymptotics of diagonal Hermite-Padé polynomials, J. Approx. Theory 42 (1984), 299-386.
  46. Papageorgiou V., Grammaticos B., Ramani A., Orthogonal polynomial approach to discrete Lax pairs for initial-boundary value problems of the QD algorithm, Lett. Math. Phys. 34 (1995), 91-101.
  47. Perron O., Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus, Math. Ann. 64 (1907), 1-76.
  48. Santini P.M., Nieszporski M., Doliwa A., Integrable generalization of the Toda law to the square lattice, Phys. Rev. E 70 (2004), 056615, 6 pages, nlin.SI/0409050.
  49. Spicer P.E., Nijhoff F.W., van der Kamp P.H., Higher analogues of the discrete-time Toda equation and the quotient-difference algorithm, Nonlinearity 24 (2011), 2229-2263, arXiv:1005.0482.
  50. Spiridonov V., Zhedanov A., Discrete Darboux transformations, the discrete-time Toda lattice, and the Askey-Wilson polynomials, Methods Appl. Anal. 2 (1995), 369-398.
  51. Spiridonov V., Zhedanov A., Spectral transformation chains and some new biorthogonal rational functions, Comm. Math. Phys. 210 (2000), 49-83.
  52. Suris Yu.B., The problem of integrable discretization: Hamiltonian approach, Progress in Mathematics, Vol. 219, Birkhäuser Verlag, Basel, 2003.
  53. Toda M., Vibration of a chain with nonlinear interaction, J. Phys. Soc. Japan 22 (1967), 431-436.
  54. Toda M., Theory of nonlinear lattices, Springer Series in Solid-State Sciences, Vol. 20, 2nd ed., Springer-Verlag, Berlin, 1989.
  55. Van Assche W., Nearest neighbor recurrence relations for multiple orthogonal polynomials, J. Approx. Theory 163 (2011), 1427-1448, arXiv:1104.3778.
  56. Van Iseghem J., Vector Stieltjes continued fraction and vector QD algorithm, Numer. Algorithms 33 (2003), 485-498.

Previous article  Next article   Contents of Volume 12 (2016)