## Long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation.(English)Zbl 1309.35147

This work provides a long-time asymptotic formula for the solutions of $i\frac{d}{dt}R_n+(R_{n+1}-2R_n+R_{n-1})-|R_n|^2(R_{n+1}+R_{n-1})=0$ which is the discrete version of the defocusing NLS (the focusing NLS is likely to be more complicated because of the soliton solutions). This equation can be exactly solved by the inverse scattering transform (IST) as done by M. J. Ablowitz et al. [Discrete and continuous nonlinear Schrödinger systems. Cambridge: Cambridge University Press (2004; Zbl 1057.35058)].
The asymptotic expansion is computed by using the so-called non-linear steepest descent method [P. A. Deift et al., in: Important developments in soliton theory. Berlin: Springer-Verlag. 181–204 (1993; Zbl 0926.35132)] which consists in applying the complex stationary phase method in the Plemelj integral associated to the (IST). In the quoted reference this has been done for the continuous NLS equations for which the contour is the real axis. Here the contour is a circle with four stationary points: $S_1=e^{-i\pi/4}A,\quad S_2=e^{-i\pi/4}\bar A,\quad S_3=-S_1,\quad S_4=-S_2$ where $$A=1/2(\sqrt{2+n/t}-i\sqrt{2-n/t})$$.
Thus, assuming $$n < 2 t$$ and that the initial datum $$R_n(0)$$ decreases fast enough with respect to $$n$$ the author proves that $R_n(t)=\frac{1}{\sqrt{t}}\sum_{j=1}^2C_je^{-i(p_jt+q_j\log(t))}+O(\log(t)/t)\qquad\text{as}\quad t\rightarrow\infty$ where $$C_j$$ and $$q_j$$ are defined in terms of the reflection coefficient that appears in the IST and depend on $$n/t$$.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q15 Riemann-Hilbert problems in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs

### Citations:

Zbl 1057.35058; Zbl 0926.35132
Full Text:

### References:

 [1] 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. · Zbl 1127.35059 [2] 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. · Zbl 0762.35001 [3] M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, Cambridge University Press, Cambridge Texts Appl. Math., 1997. · Zbl 0885.30001 [4] M. J. Ablowitz and J. F. Ladik, Nonlinear differential-difference equations, J. Math. Phys., 16 (1975), 598-603. · Zbl 0296.34062 [5] M. J. Ablowitz and J. F. Ladik, Nonlinear differential-difference equations and Fourier analysis, J. Math. Phys., 17 (1976), 1011-1018. · Zbl 0322.42014 [6] 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. · Zbl 0261.35070 [7] 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. · Zbl 1057.35058 [8] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM Stud. Appl. Math., 4 , SIAM, 1981. · Zbl 0472.35002 [9] R. Beals and R. R. Coifman, Scattering and inverse scattering for first order systems, Comm. Pure Appl. Math., 37 (1984), 39-90. · Zbl 0514.34021 [10] 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. · Zbl 1195.81048 [11] P. A. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lect. Notes Math., 3 , Courant Institute (1999); reprinted by AMS (2000). · Zbl 0997.47033 [12] 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. · Zbl 0926.35132 [13] 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. · Zbl 0771.35042 [14] 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. [15] A. S. Fokas, A Unified Approach to Boundary Value Problems, CBMS-NSF Regional Conf. Ser. in Appl. Math., 78 , SIAM, 2008. · Zbl 1181.35002 [16] 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. · Zbl 0773.35074 [17] H. Krüger and G. Teschl, Long-time asymptotics for the Toda lattice in the soliton region, Math. Z., 262 (2009), 585-602. · Zbl 1198.37104 [18] H. Krüger and G. Teschl, Long-time asymptotics of the Toda lattice for decaying initial data revisited, Rev. Math. Phys., 21 (2009), 61-109. · Zbl 1173.37057 [19] S. Lang, Differential and Riemannian Manifolds, Third ed., Grad. Texts in Math., 160 , Springer-Verlag, New York, 1995. · Zbl 0824.58003 [20] S. V. Manakov, Nonlinear Fraunhofer diffraction, Zh. Eksp. Teor. Fiz., 65 (1973), 1392-1398. (in Russian); Soviet Phys. JETP, 38 (1974), 693-696. [21] J. Michor, On the spatial asymptotics of solutions of the Ablowitz-Ladik hierarchy, Proc. Amer. Math. Soc., 138 (2010), 4249-4258. · Zbl 1209.37089 [22] 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. · Zbl 0687.34044 [23] 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. · Zbl 0489.34048 [24] H. Segur and M. J. Ablowitz, Asymptotic solutions of nonlinear evolution equations and a Painlevé transcendent, Phys. D, 3 (1981), 165-184. · Zbl 1194.35388 [25] X. Zhou, The Riemann-Hilbert problem and inverse scattering, SIAM. J. Math. Anal., 20 (1989), 966-986. · Zbl 0685.34021 [26] 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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.