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On the spatial asymptotics of solutions of the Ablowitz-Ladik hierarchy

Author: Johanna Michor
Journal: Proc. Amer. Math. Soc. 138 (2010), 4249-4258
MSC (2010): Primary 37K40, 37K15; Secondary 35Q55, 37K10
Published electronically: July 20, 2010
MathSciNet review: 2680051
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Abstract: We show that for decaying solutions of the Ablowitz-Ladik system, the leading asymptotic term is time independent. In addition, two arbitrary bounded solutions of the Ablowitz-Ladik system which are asymptotically close at the initial time stay close. All results are also derived for the associated hierarchy.

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  • 1. M. J. Ablowitz, G. Biondini, and B. Prinari, Inverse scattering transform for the integrable discrete nonlinear Schrödinger equation with nonvanishing boundary conditions, Inverse Problems 23 (2007), 1711-1758. MR 2348731 (2008f:37154)
  • 2. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991. MR 1149378 (93g:35108)
  • 3. M. J. Ablowitz and J. F. Ladik, Nonlinear differential-difference equations, J. Math. Phys. 16 (1975), 598-603. MR 0377223 (51:13396)
  • 4. -, Nonlinear differential-difference equations and Fourier analysis, J. Math. Phys. 17 (1976), 1011-1018. MR 0427867 (55:897)
  • 5. -, A nonlinear difference scheme and inverse scattering, Studies Appl. Math. 55 (1976), 213-229. MR 0471341 (57:11076)
  • 6. -, On the solution of a class of nonlinear partial difference equations, Studies Appl. Math. 57 (1977), 1-12. MR 0492975 (58:12018)
  • 7. 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. MR 2040621 (2005c:37117)
  • 8. R. Abraham, J.E. Marsden, and T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2$ ^{nd}$ ed., Springer, New York, 1988. MR 960687 (89f:58001)
  • 9. I. N. Bondareva, The Korteweg-de Vries equation in classes of increasing functions with prescribed asymptotic behaviour as $ \vert x\vert\rightarrow \infty$, Math. USSR Sb. 50 (1985), no. 1, 125-135. MR 0717670 (85h:35180)
  • 10. I. N. Bondareva and M. Shubin, Increasing asymptotic solutions of the Korteweg-de Vries equation and its higher analogues, Sov. Math. Dokl. 26 (1982), no. 3, 716-719. MR 0685832 (84k:35121)
  • 11. K. Deimling, Ordinary Differential Equations on Banach Spaces, Lecture Notes in Mathematics, 596, Springer, Berlin, 1977. MR 0463601 (57:3546)
  • 12. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, A method for solving the Korteweg-de Vries equation, Phys. Rev. Letters 19 (1967), 1095-1097.
  • 13. F. Gesztesy, H. Holden, J. Michor, and G. Teschl, Soliton Equations and Their Algebro-Geometric Solutions. Volume II: $ (1+1)$-Dimensional Discrete Models, Cambridge Studies in Advanced Mathematics, 114, Cambridge University Press, Cambridge, 2008. MR 2446594
  • 14. -, The Ablowitz-Ladik hierarchy revisited, in Methods of Spectral Analysis in Math. Physics, J. Janas et al. (eds.), 139-190, Oper. Theory Adv. Appl., 186, Birkhäuser, Basel, 2008.
  • 15. -, Local conservation laws and the Hamiltonian formalism for the Ablowitz-Ladik hierarchy, Stud. Appl. Math. 120 (2008), no. 4, 361-423. MR 2416645 (2009i:37156)
  • 16. H. Krüger and G. Teschl, Unique continuation for discrete nonlinear wave equations, arXiv:0904.0011.
  • 17. J. Michor, Inverse scattering transform for the Ablowitz-Ladik Hierarchy with quasi-periodic background, in preparation.
  • 18. G. Teschl, On the spatial asymptotics of solutions of the Toda lattice, Discrete Contin. Dyn. Sys. 273 (2010), no. 3, 1233-1239.
  • 19. V. E. Vekslerchik and V. V. Konotop, Discrete nonlinear Schrödinger equation under non-vanishing boundary conditions, Inverse Problems 8 (1992), 889-909. MR 1195946 (93j:35166)

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Additional Information

Johanna Michor
Affiliation: Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Wien, Austria – and – International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria

Keywords: Spatial asymptotics, Ablowitz–Ladik hierarchy
Received by editor(s): September 16, 2009
Published electronically: July 20, 2010
Additional Notes: This research was supported by the Austrian Science Fund (FWF) under Grant No. V120
Dedicated: Dedicated with great pleasure to Peter W. Michor on the occasion of his 60th birthday
Communicated by: Peter A. Clarkson
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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