Abstract
We obtain solutions of the discrete nonlinear Schrödinger equation with an impurity center in two ways. In the first of them, we construct the wave function as a series in a certain parameter. In the second, approximate method, we obtain the wave function in the continuum limit. We compare the obtained solutions with numerical results, and the relative accuracy of the solution in the form of a series does not exceed 10−15in order of magnitude.
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P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop, “The discrete nonlinear Schrödinger equation: A survey of recent results,” Internat. J. Modern Phys. B, 15, (2833–2900) (2001).
H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett., 81, (3383–3386) (1998).
A. Trombettoni and A. Smerzi, “Discrete solitons and breathers with dilute Bose-Einstein condensates,” Phys. Rev. Lett., 86, 2353–2356 (2001); arXiv:cond-mat/0103368v1 (2001).
F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, “Nonlinear excitations in arrays of Bose-Einstein condensates,” Phys. Rev. A., 64, (043606) (2001); arXiv:cond-mat/0106042v1 (2001).
P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations, and Physical Perspectives (Springer Tracts Mod. Phys., Vol. 232, A. Fujimori, J. Kühn, Th. Müller, F. Steiner, Trümper, and P. Wölfle, eds.), Springer, Berlin (2009).
J. C. Eilbeck and M. Johanson, “The discrete nonlinear Schrüdinger equation — 20 years on,” in: Localization and Energy Transfer in Nonlinear Systems (Madrid, Spain, 17–21 June 2002, L. Vázquez, R. S. MacKay, and M. Paz Zorzano, eds.), World Scientific, Singapore (2003), pp. 44–67.
M. J. Ablowitz, B. Prinari, and A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems (London Math. Soc. Lect. Note Ser., Vol. 302), Cambridge Univ. Press, Cambridge (2004).
R. Schärf and A. R. Bishop, “Properties of the nonlinear Schrödinger equation on a lattice,” Phys. Rev. A, 43, (6535–6544) (1991).
N. Molkenthin, S. Hu, and A. J. Niemi, “Discrete nonlinear Schrödinger equation and polygonal solitons with applications to collapsed proteins,” Phys. Rev. Lett., 106, (078102) (2011).
J. C. Eilbeck, P. S. Lomdahl, and A. C. Scott, “The discrete self-trapping equation,” Phys. D, 16, (318–338) (1985).
A. Hasegawa, Optical Solitons in Fibers, Springer, Berlin (1989).
S. V. Dmitriev, P. G. Kevrekidis, N. Yoshikawa, and D. J. Frantzeskakis, “Exact stationary solutions for the translationally invariant discrete nonlinear Schrödinger equations,” J. Phys. A: Math. Theor., 40, (1727–1746) (2007).
S. V. Dmitriev, P. G. Kevrekidis, A. A. Sukhorukov, N. Yoshikawa, and S. Takeno, “Discrete nonlinear Schrödinger equations free of the Peierls-Nabarro potential,” Phys. Lett. A, 356, (324–332) (2006).
D. E. Pelinovsky, “Translationally invariant nonlinear Schrödinger lattices,” Nonlinearity, 19, 2695–2716 (2006); arXiv:nlin/0603022v1 (2006).
D. E. Pelinovsky, T. R. O. Melvin, and A. R. Champneys, “One-parameter localized traveling waves in non-linear Schrödinger lattices,” Phys. D, 236, (22–43) (2007).
D. E. Pelinovsky and V. M. Rothos, “Bifurcations of travelling breathers in the discrete NLS equations,” Phys. D, 202, (16–36) (2005).
W.-X. Qin and X. Xiao, “Homoclinic orbits and localized solutions in nonlinear Schrödinger lattices,” Nonlinearity, 20, (2305–2317) (2007).
M. Jenkinson and M. I. Weinstein, “Onsite and offsite bound states of the discrete nonlinear Schrödinger equation and the Peierls-Nabarro barrier,” Nonlinearity, 29, (27–86) (2016).
A. Khare, K. Ø. Rasmussen, M. R. Samuelsen, and A. Saxena, “‘Exact solutions of the saturable discrete nonlinear Schrödinger equation,” J. Phys. A: Math. Gen., 38, 807–814 (2005); arXiv:nlin/0409057v1 (2004).
J. Cuevas, J. C. Eilbeck, and N. I. Karachalios, “Thresholds for breather solutions of the discrete nonlinear Schröodinger equation with saturable and power nonlinearity,” Discrete Contin. Dyn. Syst., 21, (445–475) (2008).
A. Pankov and G. Zhang, “Standing wave solutions for discrete nonlinear Schrödinger equations with unbounded potentials and saturable nonlinearity,” J. Math. Sci., 177, (71–82) (2011).
I. Aslan, “Exact and explicit solutions to the discrete nonlinear Schröodinger equation with a saturable nonlinearity,” Phys. Lett. A, 375, (4214–4217) (2011).
L. Zhang and S. Ma, “Ground state solutions for periodic discrete nonlinear Schröodinger equations with saturable nonlinearities,” Adv. Differ. Equ., 2018, (176) (2018).
D. Hennig, K. Ø. Rasmussen, H. Gabriel, and A. Bülow, “Solitonlike solutions of the generalized discrete nonlinear Schröodinger equation,” Phys.Rev.E, 54, (5788–5801) (1996).
A. Khare, K. Ø. Rasmussen, M. Salerno, M. R. Samuelsen, and A. Saxena, “Discrete nonlinear Schröodinger equations with arbitrarily high-order nonlinearities,” Phys. Rev. E, 74, (016607) (2006).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 201, No. 3, pp. 415–423, December, 2019.
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Likhachev, V.N., Vinogradov, G.A. Solutions of the Discrete Nonlinear Schrödinger Equation with a Trap. Theor Math Phys 201, 1771–1778 (2019). https://doi.org/10.1134/S0040577919120080
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DOI: https://doi.org/10.1134/S0040577919120080