Asymptotic behavior of solutions of defocusing integrable discrete nonlinear Schr?dinger equation

Abstract

We report our recent result about the long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation of Ablowitz- Ladik. The leading term is a sum of two terms that oscillate with decay of order t^-1/2

Keywords

Discrete nonlinear Schrödinger equation / Ablowitz-Ladik model / asymptotics / inverse scattering transform / nonlinear steepest descent

References

[1] Ablowitz M J, Ladik J F. Nonlinear differential-difference equations. J Math Phys, 1975, 16: 598-603

[2] Ablowitz M J, Ladik J F. Nonlinear differential-difference equations and Fourier analysis. J Math Phys, 1976, 17: 1011-1018

[3] Ablowitz M J, Newell A C. The decay of the continuous spectrum for solutions of the Korteweg-de Vries equation. J Math Phys, 1973, 14: 1277-1284

[4] Ablowitz M J, Prinari B, Trubatch A D. Discrete and Continuous Nonlinear Schrödinger Systems. Cambridge: Cambridge University Press, 2004

[5] Ablowitz M J, Biondini G, Prinari B. Inverse scattering transform for the integrable discrete nonlinear Schrödinger equation with nonvanishing boundary conditions. Inverse Problems, 2007, 231711-1758

[6] Deift P A. Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. New York:Courant Institute of Mawthematical Sciences/Providence: Amer Math Soc, 1998

[7] Deift P A, Its A R, Zhou X. Long-time asymptotics for integrable nonlinear wave equations. In: Fokas A S, Zakharov V E, eds. Important Developments in Soliton Theory, 1980-1990. Berlin: Springer-Verlag, 1993, 181-204

[8] Deift P A, Zhou X. A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann of Math (2), 1993, 137(2):295-368

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

[10] Krüger H, Teschl G. Long-time asymptotics of the Toda lattice in the soliton region. Math Z, 2009, 262(3): 585-602

[11] Krüger H, Teschl G. Long-time asymptotics of the Toda lattice for decaying initial data revisited. Rev Math Phys, 2009, 21(1): 61-109

[12] Manakov S V. Nonlinear Fraunhofer diffraction. Zh Eksp Teor Fiz, 1973, 65: 1392-1398 (in Russian); Sov Phys-JETP, 1974, 38: 693-696

[13] Michor J. On the spatial asymptotics of solutions of the Ablowitz-Ladik hierarchy. Proc Amer Math Soc, 2010, 138: 4249-4258

[14] Novokshënov V Yu. Asymptotic behavior as t → ∞ of the solution of the Cauchy problem for a nonlinear differential-difference Schrödinger equation. Differentsialnye Uravneniya, 1985, 21(11): 1915-1926 (in Russian); Differential Equations, 1985, 21(11): 1288-1298

[15] Novokshënov V Yu, Habibullin I T. Nonlinear differential-difference schemes that are integrable by the inverse scattering method. Asymptotic behavior of the solution as t → ∞. Dokl Akad Nauk SSSR, 1981, 257(3): 543-547 (in Russian); Soviet Math Dokl, 1981, 23(2): 304-308

[16] Yamane H. Long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation. arXiv: 1112.0919v2

[17] Zakharov V E, Manakov S V. Asymptotic behavior of nonlinear wave systems integrated by the inverse method. Zh Eksp Teor Fiz, 1976, 71: 203-215 (in Russian); Sov Phys-JETP, 1976, 44: 106-112

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