## Journal of the Mathematical Society of Japan

### Long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation

Hideshi YAMANE

#### Abstract

We investigate the long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation of Ablowitz-Ladik by means of the inverse scattering transform and the Deift-Zhou nonlinear steepest descent method. The leading part is a sum of two terms that oscillate with decay of order $t^{-1/2}$.

#### Article information

Source
J. Math. Soc. Japan, Volume 66, Number 3 (2014), 765-803.

Dates
First available in Project Euclid: 24 July 2014

https://projecteuclid.org/euclid.jmsj/1406206972

Digital Object Identifier
doi:10.2969/jmsj/06630765

Mathematical Reviews number (MathSciNet)
MR3238317

Zentralblatt MATH identifier
1309.35147

#### Citation

YAMANE, Hideshi. Long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation. J. Math. Soc. Japan 66 (2014), no. 3, 765--803. doi:10.2969/jmsj/06630765. https://projecteuclid.org/euclid.jmsj/1406206972

#### References

• 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), 1711–1758.
• M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Math. Soc. Lecture Note Ser., 149, Cambridge University Press, 1991.
• M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, Cambridge University Press, Cambridge Texts Appl. Math., 1997.
• M. J. Ablowitz and J. F. Ladik, Nonlinear differential-difference equations, J. Math. Phys., 16 (1975), 598–603.
• M. J. Ablowitz and J. F. Ladik, Nonlinear differential-difference equations and Fourier analysis, J. Math. Phys., 17 (1976), 1011–1018.
• M. J. Ablowitz and A. C. Newell, The decay of the continuous spectrum for solutions of the Korteweg-de Vries equation, J. Math. Phys., 14 (1973), 1277–1284.
• M. J. Ablowitz, B. Prinari and A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, London Math. Soc. Lecture Note Ser., 302, Cambridge University Press, 2004.
• M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM Stud. Appl. Math., 4, SIAM, 1981.
• R. Beals and R. R. Coifman, Scattering and inverse scattering for first order systems, Comm. Pure Appl. Math., 37 (1984), 39–90.
• K. W. Chow, Robert Conte and Neil Xu, Analytic doubly periodic wave patterns for the integrable discrete nonlinear Schrödinger (Ablowitz-Ladik) model, Phys. Lett. A, 349 (2006), 422–429.
• P. A. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lect. Notes Math., 3, Courant Institute (1999); reprinted by AMS (2000).
• P. A. Deift, A. R. Its and X. Zhou, Long-time asymptotics for integrable nonlinear wave equations, In: Important Developments in Soliton Theory, 1980–1990 (eds. A. S. Fokas and V. E. Zakharov), Springer Ser. Nonlinear Dynam., Springer-Verlag, 1993, pp.,181–204.
• P. A. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2), 137 (1993), 295–368.
• P. A. Deift and X. Zhou, Long-time behavior of the non-focusing nonlinear Schrödinger equation – a case study, Lectures in Mathematical Sciences, No.,5, The University of Tokyo, 1994.
• A. S. Fokas, A Unified Approach to Boundary Value Problems, CBMS-NSF Regional Conf. Ser. in Appl. Math., 78, SIAM, 2008.
• S. Kamvissis, On the long time behavior of the doubly infinite Toda lattice under initial data decaying at infinity, Comm. Math. Phys., 153 (1993), 479–519.
• H. Krüger and G. Teschl, Long-time asymptotics for the Toda lattice in the soliton region, Math. Z., 262 (2009), 585–602.
• H. Krüger and G. Teschl, Long-time asymptotics of the Toda lattice for decaying initial data revisited, Rev. Math. Phys., 21 (2009), 61–109.
• S. Lang, Differential and Riemannian Manifolds, Third ed., Grad. Texts in Math., 160, Springer-Verlag, New York, 1995.
• S. V. Manakov, Nonlinear Fraunhofer diffraction, Zh. Eksp. Teor. Fiz., 65 (1973), 1392–1398. (in Russian); Soviet Phys. JETP, 38 (1974), 693–696.
• J. Michor, On the spatial asymptotics of solutions of the Ablowitz-Ladik hierarchy, Proc. Amer. Math. Soc., 138 (2010), 4249–4258.
• V. Yu. Novokshënov, Asymptotic behavior as $t\to\infty$ of the solution of the Cauchy problem for a nonlinear differential-difference Schrödinger equation, Differentsial'nye Uravneniya, 21 (1985), 1915–1926. (in Russian); Differential Equations, 21 (1985), 1288–1298.
• V. Yu. Novokshënov and I. T. Habibullin, Nonlinear differential-difference schemes that are integrable by the inverse scattering method. Asymptotic behavior of the solution as $t\to\infty$, Dokl. Akad. Nauk SSSR, 257 (1981), 543–547. (in Russian); Soviet Math. Dokl., 23 (1981), 304–308.
• H. Segur and M. J. Ablowitz, Asymptotic solutions of nonlinear evolution equations and a Painlevé transcendent, Phys. D, 3 (1981), 165–184.
• X. Zhou, The Riemann-Hilbert problem and inverse scattering, SIAM. J. Math. Anal., 20 (1989), 966–986.
• V. E. Zakharov and S. V. Manakov, Asymptotic behavior of non-linear wave systems integrated by the inverse scattering method, Z. Èksper. Teoret. Fiz., 71 (1976), 203–215. (in Russian); Soviet Physics JETP, 44 (1976), 106–112.