Abstract
We derive theta function representations of algebro-geometric solutions of a discrete system governed by a transfer matrix associated with (an extension of) the trigonometric moment problem studied by Szegő and Baxter. We also derive a new hierarchy of coupled nonlinear difference equations satisfied by these algebro-geometric solutions.
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Ablowitz, M.J.: Nonlinear evolution equations – continuous and discrete. SIAM Rev. 19, 663–684 (1977)
Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge: Cambridge University Press, 1991
Ablowitz, M.J., Ladik, J.F.: Nonlinear differential-difference equations. J. Math. Phys. 16, 598–603 (1975)
Ablowitz, M.J., Ladik, J.F.: Nonlinear differential-difference equations and Fourier analysis J. Math. Phys. 17, 1011–1018 (1976)
Ablowitz, M.J., Ladik, J.F.: A nonlinear difference scheme and inverse scattering. Studies Appl. Math. 55, 213–229 (1976)
Ablowitz, M.J., Ladik, J.F.: On the solution of a class of nonlinear partial difference equations. Studies Appl. Math. 57, 1–12 (1977)
Ablowitz, M.J., Prinari, B., Trubatch, A.D.: Discrete and Continuous Nonlinear Schrödinger Systems. London Mathematical Society Lecture Note Series, Vol. 302, Cambridge: Cambridge University Press, 2004
Ahmad, S., Roy Chowdhury, A.: On the quasi-periodic solutions to the discrete non-linear Schrödinger equation. J. Phys. A 20, 293–303 (1987)
Akhiezer, N.I.: The Classical Moment Problem. Edinburgh: Oliver & Boyd., 1965
Baxter, G.: Polynomials defined by a difference system. Bull. Amer. Math. Soc. 66, 187–190 (1960)
Baxter, G.: Polynomials defined by a difference system. J. Math. Anal. Appl. 2, 223–263 (1961)
Baxter, G.: A convergence equivalence related to polynomials orthogonal on the unit circle. Trans. Amer. Math. Soc. 99, 471–487 (1961)
Baxter, G.: A norm inequality for a “finite-section” Wiener-Hopf equation. Illinois J. Math. 7, 97–103 (1963)
Belokolos, E.D., Bobenko, A.I., Enol’skii, V.Z., Its, A.R., Matveev, V.B.: Algebro-Geometric Approach to Nonlinear Integrable Equations. Berlin: Springer, 1994
Bogolyubov, N.N., Prikarpatskii, A.K., Samoilenko, V.G.: Discrete periodic problem for the modified nonlinear Korteweg–de Vries equation. Sov. Phys. Dokl. 26, 490–492 (1981)
Bogolyubov, N.N., Prikarpatskii, A.K.: The inverse periodic problem for a discrete approximation of a nonlinear Schrödinger equation. Sov. Phys. Dokl. 27, 113–116 (1982)
Bulla, W., Gesztesy, F., Holden, H., Teschl, G.: Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchy. Mem. Amer. Math. Soc. no. 641, 135, 1–79 (1998)
Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes, Vol. 3. Providence, R.I.: Courant Institute of Mathematical Sciences, New York University and Amer. Math. Soc., 2002
Farkas, H.M., Kra, I.: Riemann Surfaces. 2nd ed., New York: Springer, 1992
Faybusovich, L., Gekhtman, M.: On Schur flows. J. Phys. A 32, 4671–4680 (1999)
Geng, X., Dai, H.H., Cao, C.: Algebro-geometric constructions of the discrete Ablowitz–Ladik flows and applications. J. Math. Phys. 44, 4573–4588 (2003)
Geronimo, J.S., Johnson, R.: Rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle. J. Differ. Eqs. 132, 140–178 (1996)
Geronimo, J.S., Johnson, R.: An inverse problem associated with polynomials orthogonal on the unit circle. Commun. Math. Phys. 193, 125–150 (1998)
Geronimo, J.S., Teplyaev, A.: A difference equation arising from the trigonometric moment problem having random reflection coefficients–an operator theoretic approach. J. Funct. Anal. 123, 12–45 (1994)
Geronimus, J.: On the trigonometric moment problem. Ann. Math. 47, 742–761 (1946)
Geronimus, Ya.L.: Polynomials orthogonal on a circle and their applications. Commun. Soc. Mat. Kharkov 15, 35–120 (1948); Amer. Math. Soc. Transl. (1) 3:1–78, (1962)
Geronimus, Ya.L.: Orthogonal Polynomials. New York: Consultants Bureau, 1961
Gesztesy, F., Holden, H.: Soliton Equations and Their Algebro-Geometric Solutions. Volume I: (1+1)-Dimensional Continuous Models. Cambridge Studies in Advanced Mathematics, Vol. 79, Cambridge: Cambridge University Press, 2003
Gesztesy, F., Holden, H.: Soliton Equations and Their Algebro-Geometric Solutions. Volume II: (1+1)-Dimensional Discrete Models. Cambridge Studies in Advanced Mathematics, Cambridge: Cambridge University Press, in preparation
Gesztesy, F., Ratnaseelan, R.: An alternative approach to algebro-geometric solutions of the AKNS hierarchy. Rev. Math. Phys. 10, 345–391 (1998)
Gesztesy, F., Ratnaseelan, R., Teschl, G.: The KdV hierarchy and associated trace formulas. In I. Gohberg, P. Lancaster, P. N. Shivakumar, eds., Recent Developments in Operator Theory and Its Applications, Volume 87 of Operator Theory: Advances and Applications, Basel: Birkhäuser, 1996, pp. 125–163
Gesztesy, F., Zinchenko, M.: A Borg-type theorem associated with orthogonal polynomials on the unit circle. Preprint, 2004
Grenander, U., Szegő, G.: Toeplitz Forms and their Applications. Berkeley, CA: University of California Press, 1958; 2nd ed., New York: Chelsea, 1984
Krein, M.G.: On a generalization of some investigations of G. Szegő, V. Smirnoff, and A. Kolmogoroff. Doklady Akad. Nauk SSSR 46, 91–94 1945 (Russian)
Miller, P.D., Ercolani, N.M., Krichever, I.M., Levermore, C.D.: Finite genus solutions to the Ablowitz–Ladik equations. Comm. Pure Appl. Math. 48, 1369–1440 (1995)
Mukaihira, A., Nakamura, Y.: Schur flow for orthogonal polynomials on the unit circle and its integrable discretization. J. Comput. Appl. Math. 139, 75–94 (2002)
Mumford, D.: Tata Lectures on Theta II. Boston: Birkhäuser, 1984
Nenciu, I.: Lax pairs for the Ablowitz-Ladik system via orthogonal polynomials on the unit circle. Intl. Math. Res. Notices, to appear
Peherstorfer, F.: In preparation
Simon, B.: Analogs of the m-function in the theory of orthogonal polynomials on the unit circle. J. Comp. Appl. Math. 171, 411–424 (2004)
Simon, B.: Orthogonal Polynomials on the Unit Circle, Vols. 1 and 2. AMS Colloquium Publication Series, Providence, R.I.: Amer. Math. Soc., 2005
Simon, B.: Orthogonal polynomials on the unit circle: New results. Intl. Math. Res. Notices 2004, No. 53, 2837–2880
Szegő, G.: Beiträge zur Theorie der Toeplitzschen Formen I. Math. Z. 6, 167–202 (1920)
Szegő, G.: Beiträge zur Theorie der Toeplitzschen Formen II. Math. Z. 9, 167–190 (1921)
Szegő, G.: Orthogonal Polynomials. Amer Math. Soc. Colloq. Publ. Vol. 23, Providence, R.I.: Amer. Math. Soc., 1978
Tomčuk, Ju.Ja.: Orthogonal polynomials on a given system of arcs of the unit circle. Sov. Math. Dokl. 4, 931–934 (1963)
Vekslerchik, V.E.: Finite genus solutions for the Ablowitz–Ladik hierarchy. J. Phys. A 32:4983–4994, (1998)
Verblunsky, S.: On positive harmonic functions: A contribution to the algebra of Fourier series. Proc. London Math. Soc. (2) 38, 125–157 (1935)
Verblunsky, S.: On positive harmonic functions (second paper). Proc. London Math. Soc. (2) 40, 290–320 (1936)
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Communicated by B. Simon
Supported in part by the US National Science Foundation under Grants No. DMS-0200219 and DMS-0405526.
The research of the second and third author was supported in part by the Research Council of Norway
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Geronimo, J., Gesztesy, F. & Holden, H. Algebro-Geometric Solutions of the Baxter–Szegő Difference Equation. Commun. Math. Phys. 258, 149–177 (2005). https://doi.org/10.1007/s00220-005-1305-x
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DOI: https://doi.org/10.1007/s00220-005-1305-x