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

Michor, Johanna

doi:10.1090/S0002-9939-2010-10595-6

On the spatial asymptotics of solutions of the Ablowitz–Ladik hierarchy
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Proc. Amer. Math. Soc. 138 (2010), 4249-4258 Request permission

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.

References

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, DOI 10.1088/0266-5611/23/4/021
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, DOI 10.1017/CBO9780511623998
M. J. Ablowitz and J. F. Ladik, Nonlinear differential-difference equations, J. Mathematical Phys. 16 (1975), 598–603. MR 377223, DOI 10.1063/1.522558
M. J. Ablowitz and J. F. Ladik, Nonlinear differential-difference equations and Fourier analysis, J. Mathematical Phys. 17 (1976), no. 6, 1011–1018. MR 427867, DOI 10.1063/1.523009
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 471341, DOI 10.1002/sapm1976553213
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 492975, DOI 10.1002/sapm19775711
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
R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, tensor analysis, and applications, 2nd ed., Applied Mathematical Sciences, vol. 75, Springer-Verlag, New York, 1988. MR 960687, DOI 10.1007/978-1-4612-1029-0
I. N. Bondareva, The Korteweg-de Vries equation in classes of growing functions with a given asymptotic behavior as $\mid x\mid \rightarrow \infty$, Mat. Sb. (N.S.) 122(164) (1983), no. 2, 131–141 (Russian). MR 717670
I. N. Bondareva and M. A. Shubin, Growing asymptotic solutions of the Korteweg-de Vries equation and of its higher analogues, Dokl. Akad. Nauk SSSR 267 (1982), no. 5, 1035–1038 (Russian). MR 685832
Klaus Deimling, Ordinary differential equations in Banach spaces, Lecture Notes in Mathematics, Vol. 596, Springer-Verlag, Berlin-New York, 1977. MR 0463601, DOI 10.1007/BFb0091636
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.
Fritz Gesztesy, Helge Holden, Johanna Michor, and Gerald Teschl, Soliton equations and their algebro-geometric solutions. Vol. II, Cambridge Studies in Advanced Mathematics, vol. 114, Cambridge University Press, Cambridge, 2008. $(1+1)$-dimensional discrete models. MR 2446594, DOI 10.1017/CBO9780511543203
—, 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.
Fritz Gesztesy, Helge Holden, Johanna Michor, and Gerald Teschl, Local conservation laws and the Hamiltonian formalism for the Ablowitz-Ladik hierarchy, Stud. Appl. Math. 120 (2008), no. 4, 361–423. MR 2416645, DOI 10.1111/j.1467-9590.2008.00405.x
H. Krüger and G. Teschl, Unique continuation for discrete nonlinear wave equations, arXiv:0904.0011.
J. Michor, Inverse scattering transform for the Ablowitz–Ladik Hierarchy with quasi-periodic background, in preparation.
G. Teschl, On the spatial asymptotics of solutions of the Toda lattice, Discrete Contin. Dyn. Sys. 273 (2010), no. 3, 1233–1239.
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, DOI 10.1088/0266-5611/8/6/007