Lax pairs for the Ablowitz-Ladik system via orthogonal polynomials on the unit circle

International Mathematics Research Notices

Volumn , Issue 11, 2005, Pages 647-686

  Nenciu, Irina ^a  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 17744367851     PISSN: 10737928     EISSN: None     Source Type: Journal    
DOI: 10.1155/imrn.2005.647     Document Type: Article

References (18)

1. M. J. Ablowitz and J. F. Ladik, Nonlinear differential-difference equations, J. Math. Phys. 16 (1975), 598-603.
2. _, Nonlinear differential-difference equations and Fourier analysis, J. Math. Phys. 17 (1976), no. 6, 1011-1018.
3. M. J. Ablowitz, B. Prinari, and A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, London Mathematical Society Lecture Note Series, vol. 302, Cambridge University Press, Cambridge, 2004.
4. V. I. Arnol'd, V. V. Kozlov, and A. I. Neǐshtadt, Mathematical aspects of classical and celestial mechanics, Dynamical Systems, III, Encyclopaedia Math. Sci., vol. 3, Springer, Berlin, 1993, pp. vii-xiv and 1-291.
5. M. J. Cantero, L. Moral, and L. Velázquez, Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle, Linear Algebra Appl. 362 (2003), 29-56.
6. _, Minimal representations of unitary operators and orthogonal polynomials on the unit circle, preprint, 2004, http://arxiv.org/math.CA/ 0405246.
7. P. Deift, Integrable Hamiltonian systems, Dynamical Systems and Probabilistic Methods in Partial Differential Equations (Berkeley, Calif, 1994), Lectures in Appl. Math., vol. 31, American Mathematical Society, Rhode Island, 1996, pp. 103-138.
8. J. Geronimo, F. Gesztesy, and H. Holden, Algebro-geometric solutions of a discrete system related to the trigonometric moment problem, preprint, 2004, http://arxiv.org/math.SP/0408073.
9. F. Gesztesy and H. Holden, Soliton Equations and Their Algebro-Geometric Solutions. Vol. II: (1 + 1)-Dimensional Discrete Models, Cambridge Studies in Adv. Math., Cambridge University Press, Cambridge, in press.
10. P. D. Miller, N. M. Ercolani, I. M. Krichever, and C. D. Levermore, Finite genus solutions to the Ablowitz-Ladik equations, Comm. Pure Appl. Math. 48 (1995), no. 12, 1369-1440.
11. P. D. Miller, A. C. Scott, J. Carr, and J. C. Eilbeck, Binding energies for discrete nonlinear Schrödinger equations, Phys. Scripta 44 (1991), no. 6, 509-516.
12. D. E. Rourke, Elementary Bäcklund transformations for a discrete Ablowitz-Ladik eigenvalue problem, J. Phys. A 37 (2004), no. 7, 2693-2708.
13. B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Rhode Island, 2005.
14. _, Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Rhode Island, 2005.
15. G. Szegö, Orthogonal Polynomials, 4th ed., American Mathematical Society Colloquium Publications, vol. 23, American Mathematical Society, Rhode Island, 1975.
16. P. van Moerbeke, The spectrum of Jacobi matrices, Invent. Math. 37 (1976), no. 1, 45-81.
17. K. L. Vaninsky, Symplectic structures and volume elements in the function space for the cubic Schrödinger equation, Duke Math. J. 92 (1998), no. 2, 381-402.
18. _, An additional Gibbs' state for the cubic Schrödinger equation on the circle, Comm. Pure Appl. Math. 54 (2001), no. 5, 537-582.