Geodesic
Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We investigate the long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation. If $|n|2t$, we have decaying oscillation of order $O(t^{-1/2})$ as was proved in our previous paper. Near $|n|=2t$, the behavior is decaying oscillation of order $O(t^{-1/3})$ and the coefficient of the leading term is expressed by the Painlevé II function. In $|n|>2t$, the solution decays more rapidly than any negative power of $n$.
Keywords: discrete nonlinear Schrödinger equation; nonlinear steepest descent; Painlevé equation. 
@article{SIGMA_2015_11_a19,
     author = {Hideshi Yamane},
     title = {Long-Time {Asymptotics} for the {Defocusing} {Integrable} {Discrete} {Nonlinear} {Schr\"odinger} {Equation~II}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2015},
     volume = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a19/}
}
TY  - JOUR
AU  - Hideshi Yamane
TI  - Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2015
VL  - 11
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a19/
LA  - en
ID  - SIGMA_2015_11_a19
ER  - 
%0 Journal Article
%A Hideshi Yamane
%T Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II
%J Symmetry, integrability and geometry: methods and applications
%D 2015
%V 11
%U http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a19/
%G en
%F SIGMA_2015_11_a19
Hideshi Yamane. Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a19/

[1] Ablowitz M. J., Prinari B., Trubatch A. D., Discrete and continuous nonlinear Schrödinger systems, London Mathematical Society Lecture Note Series, 302, Cambridge University Press, Cambridge, 2004

[2] Deift P., Zhou X., “A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation”, Ann. of Math., 137 (1993), 295–368, arXiv: math.AP/9201261 | DOI

[3] Kamvissis S., “On the long time behavior of the doubly infinite Toda lattice under initial data decaying at infinity”, Comm. Math. Phys., 153 (1993), 479–519 | DOI

[4] Kitaev A. V., “Caustics in $1+1$ integrable systems”, J. Math. Phys., 35 (1994), 2934–2954 | DOI

[5] Novokshenov V. Yu., “Asymptotic behavior as $t\to\infty$ of the solution of the Cauchy problem for a nonlinear differential-difference Schrödinger equation”, Differ. Equ., 21 (1985), 1288–1298

[6] Yamane H., “Long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation”, J. Math. Soc. Japan, 66 (2014), 765–803, arXiv: 1112.0919 | DOI