
SIGMA 11 (2015), 020, 17 pages arXiv:1407.5751
https://doi.org/10.3842/SIGMA.2015.020
LongTime Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II
Hideshi Yamane
Department of Mathematical Sciences, Kwansei Gakuin University, Gakuen 21 Sanda, Hyogo 6691337, Japan
Received September 06, 2014, in final form March 03, 2015; Published online March 08, 2015
Abstract
We investigate the longtime 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 $n$.
Key words:
discrete nonlinear Schrödinger equation; nonlinear steepest descent; Painlevé equation.
pdf (635 kb)
tex (245 kb)
References

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

Deift P., Zhou X., A steepest descent method for oscillatory RiemannHilbert problems. Asymptotics for the MKdV equation, Ann. of Math. 137 (1993), 295368, math.AP/9201261.

Kamvissis S., On the long time behavior of the doubly infinite Toda lattice under initial data decaying at infinity, Comm. Math. Phys. 153 (1993), 479519.

Kitaev A.V., Caustics in $1+1$ integrable systems, J. Math. Phys. 35 (1994), 29342954.

Novokshenov V.Yu., Asymptotic behavior as $t\to\infty$ of the solution of the Cauchy problem for a nonlinear differentialdifference Schrödinger equation, Differ. Equ. 21 (1985), 12881298.

Yamane H., Longtime asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation, J. Math. Soc. Japan 66 (2014), 765803, arXiv:1112.0919.

