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
Fulltext PDF
Abstract 
References 
Similar Articles 
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.
 1.
Mark
J. Ablowitz, Gino
Biondini, and Barbara
Prinari, Inverse scattering transform for the integrable discrete
nonlinear Schrödinger equation with nonvanishing boundary
conditions, Inverse Problems 23 (2007), no. 4,
1711–1758. MR 2348731
(2008f:37154), http://dx.doi.org/10.1088/02665611/23/4/021
 2.
M.
J. Ablowitz and P.
A. Clarkson, Solitons, nonlinear evolution equations and inverse
scattering, London Mathematical Society Lecture Note Series,
vol. 149, Cambridge University Press, Cambridge, 1991. MR 1149378
(93g:35108)
 3.
M.
J. Ablowitz and J.
F. Ladik, Nonlinear differentialdifference equations, J.
Mathematical Phys. 16 (1975), 598–603. MR 0377223
(51 #13396)
 4.
M.
J. Ablowitz and J.
F. Ladik, Nonlinear differentialdifference equations and Fourier
analysis, J. Mathematical Phys. 17 (1976),
no. 6, 1011–1018. MR 0427867
(55 #897)
 5.
M.
J. Ablowitz and J.
F. Ladik, A nonlinear difference scheme and inverse
scattering, Studies in Appl. Math. 55 (1976),
no. 3, 213–229. MR 0471341
(57 #11076)
 6.
M.
J. Ablowitz and J.
F. Ladik, On the solution of a class of nonlinear partial
difference equations, Studies in Appl. Math. 57
(1976/77), no. 1, 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, 2nd ed.,
Applied Mathematical Sciences, vol. 75, SpringerVerlag, New York,
1988. MR
960687 (89f:58001)
 9.
I.
N. Bondareva, The Kortewegde Vries equation in classes of growing
functions with a given asymptotic behavior as
\mid𝑥\mid→∞, Mat. Sb. (N.S.)
122(164) (1983), no. 2, 131–141 (Russian). MR 717670
(85h:35180)
 10.
I.
N. Bondareva and M.
A. Shubin, Growing asymptotic solutions of the Kortewegde Vries
equation and of its higher analogues, Dokl. Akad. Nauk SSSR
267 (1982), no. 5, 1035–1038 (Russian). MR 685832
(84k:35121)
 11.
Klaus
Deimling, Ordinary differential equations in Banach spaces,
Lecture Notes in Mathematics, Vol. 596, SpringerVerlag, BerlinNew York,
1977. MR
0463601 (57 #3546)
 12.
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.
 13.
Fritz
Gesztesy, Helge
Holden, Johanna
Michor, and Gerald
Teschl, Soliton equations and their algebrogeometric solutions.
Vol. II, Cambridge Studies in Advanced Mathematics, vol. 114,
Cambridge University Press, Cambridge, 2008. (1+1)dimensional discrete
models. MR
2446594 (2011b:37130)
 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.
Fritz
Gesztesy, Helge
Holden, Johanna
Michor, and Gerald
Teschl, Local conservation laws and the Hamiltonian formalism for
the AblowitzLadik hierarchy, Stud. Appl. Math. 120
(2008), no. 4, 361–423. MR 2416645
(2009i:37156), http://dx.doi.org/10.1111/j.14679590.2008.00405.x
 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.
Vadim
E. Vekslerchik and Vladimir
V. Konotop, Discrete nonlinear Schrödinger equation under
nonvanishing boundary conditions, Inverse Problems 8
(1992), no. 6, 889–909. MR 1195946
(93j:35166)
 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), 17111758. 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 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)
 9.
 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)
 10.
 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)
 12.
 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.
 13.
 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)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
37K40,
37K15,
35Q55,
37K10
Retrieve articles in all journals
with MSC (2010):
37K40,
37K15,
35Q55,
37K10
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.
