Proceedings of the American Mathematical Society

Inverse scattering transform for the Toda hierarchy with quasi-periodic background

Author(s): Iryna Egorova; Johanna Michor; Gerald Teschl
Journal: Proc. Amer. Math. Soc. 135 (2007), 1817-1827.
MSC (2000): Primary 37K15, 37K10; Secondary 47B36, 34L25
Posted: November 7, 2006
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Abstract: We provide a rigorous treatment of the inverse scattering transform for the entire Toda hierarchy in the case of a quasi-periodic finite-gap background solution.

1.: A. Boutet de Monvel and I. Egorova, The Toda lattice with step-like initial data. Soliton asymptotics, Inverse Problems 16, No. 4, 955-977 (2000). MR 1776477 (2001g:37138)
2.: W. Bulla, F. Gesztesy, H. Holden, and G. Teschl, Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies, Memoirs of the Amer. Math. Soc. 135/641 (1998). MR 1432141 (99b:58109)
3.: I. Egorova, J. Michor, and G. Teschl, Scattering theory for Jacobi operators with quasi-periodic background, Comm. Math. Phys. 264-3, 811-842 (2006).
4.: L. Faddeev and L. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer, Berlin, 1987. MR 0905674 (89m:58103)
5.: N. E. Firsova, On the solution of the Cauchy problem for the Korteweg-de Vries equation with initial data being the sum of periodic and rapidly decreasing functions, Mat. Sb., N. Ser. 135(177), No. 2, 261-268 (1988).MR 0937811 (90a:35197)
6.: H. Flaschka, On the Toda lattice. II, Progr. Theoret. Phys. 51, 703-716 (1974).MR 0408648 (53:12412)
7.: C. S. Gardner, J. M. Green, M. D. Kruskal, and R. M. Miura, A method for solving the Korteweg-de Vries equation, Phys. Rev. Letters 19, 1095-1097 (1967).
8.: E. A. Kuznetsov and A. V. Mikha ${\u{\i\/}}\kern.15em$ lov, Stability of stationary waves in nonlinear weakly dispersive media, Soviet Phys. JETP 40, No. 5, 855-859 (1975).MR 0387847 (52:8685)
9.: P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure and Appl. Math. 21, 467-490 (1968). MR 0235310 (38:3620)
10.: V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser, Basel, 1986. MR 0897106 (88f:34034)
11.: J. Michor and G. Teschl, Trace formulas for Jacobi operators in connection with scattering theory for quasi-periodic background, Proc. Operator Theory and Applications in Mathematical Physics 2004, J. Janas, et al. (eds.), Oper. Theory Adv. Appl., Birkhäuser, Basel (to appear).
12.: G. Teschl, Inverse scattering transform for the Toda hierarchy, Math. Nach. 202, 163-171 (1999). MR 1694723 (2000e:37124)
13.: G. Teschl, On the Toda and Kac-van Moerbeke hierarchies, Math. Z. 231, 325-344 (1999). MR 1703351 (2000f:37105)
14.: G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surv. and Mon. 72, Amer. Math. Soc., Rhode Island, 2000.MR 1711536 (2001b:39019)
15.: M. Toda, Theory of Nonlinear Lattices, 2nd enl. ed., Springer, Berlin, 1989.MR 0971987 (89h:58082)
16.: A. Volberg and P. Yuditskii, On the inverse scattering problem for Jacobi Matrices with the Spectrum on an Interval, a finite systems of intervals or a Cantor set of positive length, Commun. Math. Phys. 226, 567-605 (2002).MR 1896882 (2003m:47059)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37K15, 37K10, 47B36, 34L25

Iryna Egorova
Affiliation: Kharkiv National University, 47 Lenin ave, 61164 Kharkiv, Ukraine
Email: egorova@ilt.kharkov.ua

Johanna Michor
Affiliation: Faculty of Mathematics, Nordbergstrasse 15, 1090 Wien, Austria -- and -- International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria
Email: Johanna.Michor@esi.ac.at

Gerald Teschl
Affiliation: Faculty of Mathematics, Nordbergstrasse 15, 1090 Wien, Austria -- and -- International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria
Email: Gerald.Teschl@univie.ac.at

DOI: 10.1090/S0002-9939-06-08668-0
PII: S 0002-9939(06)08668-0
Keywords: Inverse scattering, Toda hierarchy, periodic
Received by editor(s): December 1, 2005
Received by editor(s) in revised form: February 7, 2006
Posted: November 7, 2006
Additional Notes: This work was supported by the Austrian Science Fund (FWF) under Grant No. P17762 and INTAS Research Network NeCCA 03-51-6637.
Communicated by: Joseph A. Ball
Copyright of article: Copyright 2006, American Mathematical Society

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