Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 020, 17 pages      arXiv:1407.5751

Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II

Hideshi Yamane
Department of Mathematical Sciences, Kwansei Gakuin University, Gakuen 2-1 Sanda, Hyogo 669-1337, Japan

Received September 06, 2014, in final form March 03, 2015; Published online March 08, 2015

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 $n$.

Key words: discrete nonlinear Schrödinger equation; nonlinear steepest descent; Painlevé equation.

pdf (635 kb)   tex (245 kb)


  1. 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.
  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, math.AP/9201261.
  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.
  4. Kitaev A.V., Caustics in $1+1$ integrable systems, J. Math. Phys. 35 (1994), 2934-2954.
  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.

Previous article  Next article   Contents of Volume 11 (2015)