On the spatial asymptotics of solutions of the AblowitzLadik hierarchy
Author:
Johanna Michor
Journal:
Proc. Amer. Math. Soc. 138 (2010), 42494258
MSC (2010):
Primary 37K40, 37K15; Secondary 35Q55, 37K10
Published electronically:
July 20, 2010
MathSciNet review:
2680051
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Additional Information
Abstract: We show that for decaying solutions of the AblowitzLadik system, the leading asymptotic term is time independent. In addition, two arbitrary bounded solutions of the AblowitzLadik system which are asymptotically close at the initial time stay close. All results are also derived for the associated hierarchy.
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 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), 17111758. MR 2348731 (2008f:37154)
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 M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991. MR 1149378 (93g:35108)
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 M. J. Ablowitz and J. F. Ladik, Nonlinear differentialdifference equations, J. Math. Phys. 16 (1975), 598603. MR 0377223 (51:13396)
 4.
 , Nonlinear differentialdifference equations and Fourier analysis, J. Math. Phys. 17 (1976), 10111018. MR 0427867 (55:897)
 5.
 , A nonlinear difference scheme and inverse scattering, Studies Appl. Math. 55 (1976), 213229. MR 0471341 (57:11076)
 6.
 , On the solution of a class of nonlinear partial difference equations, Studies Appl. Math. 57 (1977), 112. 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 ed., Springer, New York, 1988. MR 960687 (89f:58001)
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 I. N. Bondareva, The Kortewegde Vries equation in classes of increasing functions with prescribed asymptotic behaviour as , Math. USSR Sb. 50 (1985), no. 1, 125135. MR 0717670 (85h:35180)
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 I. N. Bondareva and M. Shubin, Increasing asymptotic solutions of the Kortewegde Vries equation and its higher analogues, Sov. Math. Dokl. 26 (1982), no. 3, 716719. 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)
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 C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, A method for solving the Kortewegde Vries equation, Phys. Rev. Letters 19 (1967), 10951097.
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 F. Gesztesy, H. Holden, J. Michor, and G. Teschl, Soliton Equations and Their AlgebroGeometric Solutions. Volume II: Dimensional Discrete Models, Cambridge Studies in Advanced Mathematics, 114, Cambridge University Press, Cambridge, 2008. MR 2446594
 14.
 , The AblowitzLadik hierarchy revisited, in Methods of Spectral Analysis in Math. Physics, J. Janas et al. (eds.), 139190, Oper. Theory Adv. Appl., 186, Birkhäuser, Basel, 2008.
 15.
 , Local conservation laws and the Hamiltonian formalism for the AblowitzLadik hierarchy, Stud. Appl. Math. 120 (2008), no. 4, 361423. 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 AblowitzLadik Hierarchy with quasiperiodic background, in preparation.
 18.
 G. Teschl, On the spatial asymptotics of solutions of the Toda lattice, Discrete Contin. Dyn. Sys. 273 (2010), no. 3, 12331239.
 19.
 V. E. Vekslerchik and V. V. Konotop, Discrete nonlinear Schrödinger equation under nonvanishing boundary conditions, Inverse Problems 8 (1992), 889909. 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
Email:
Johanna.Michor@univie.ac.at
DOI:
http://dx.doi.org/10.1090/S000299392010105956
PII:
S 00029939(2010)105956
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.
