Abstract
A “transparent point” is a particular value of a governing parameter in a nontranslationally invariant system that makes the system “almost” translationally invariant. This concept was introduced recently in the context of the discrete nonlinear Schrödinger (DNLS) equation with saturable nonlinearity — it was discovered that a tuning of the lattice spacing parameter h in this model affects the soliton mobility. In this paper, we study the DNLS equation with competing cubic–quintic nonlinearity that also admits the transparent points with respect to the lattice spacing parameter h. We give a geometrical interpretation of the transparent points in terms of dynamical system theory and present a simple asymptotical formula for them at h → 0. Although the derivation of this formula is heuristic and nonrigorous, it gives the values of transparent points with remarkable accuracy even for quite large values of h.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. N. Christodoulides and R. I. Joseph, Opt. Lett., 13, 794 (1988).
F. Lederer, G. I. Stegeman, D. N. Christodoulides, et al., Phys. Rep., 463, 1 (2008).
A. Smerzi, A. Trombettoni, P. G. Kevrekidis, and A. R. Bishop, Phys. Rev. Lett., 89, 170402 (2002).
P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations, and Physical Perspectives, Springer (2009), STMP 232.
D. E. Pelinovsky, Nonlinearity, 19, 2695 (2006).
S. V. Dmitriev, P. G. Kevrekidis, N. Yoshikawa, and D. J. Frantzeskakis, J. Phys. A: Math. Theor., 40, 1727 (2007).
S. V. Dmitriev, P. G. Kevrekidis, A. A. Sukhorukov, et al., Phys. Lett. A, 356, 324 (2006).
V. L. Vinetskii and N. V. Kukhtarev, Fiz. Tverd. Tela, 16, 3714 (1974)) [English translation: Sov. Phys. Solid State, 16, 2414 (1975)].
S. Gatz and J. Herrmann, J. Opt. Soc. Am. B, 8, 2296 (1991).
L. Hadzievski, A. Maluckov, M. Stepic, and D. Kip, Phys. Rev. Lett., 93, 033901 (2004).
J. Cuevas and J. C. Eilbeck, Phys. Lett. A, 358, 15 (2006).
U. Naether, R. A. Vicencio, and M. Stepic, Opt. Lett., 36, 1467 (2011).
T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis, and J. Cuevas, Phys. Rev. Lett., 97, 124101 (2006).
T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis, and J. Cuevas, Physica D, 237, 551 (2008).
O. F. Oxtoby and I. V. Barashenkov, Phys. Rev. E, 76, 036603 (2007).
G. L. Alfimov, A. S. Korobeinikov, C. Lustri, and D. E. Pelinovsky, Nonlinearity, 32, 3445 (2019).
G. L. Alfimov, E. V. Medvedeva, and D. E. Pelinovsky, Phys. Rev. Lett., 112, 054103 (2014).
C. Mejía-Cortés, R. A. Vicencio, and B. A. Malomed, Phys. Rev. E, 88, 052901 (2013).
R. Carretero-Gonzlez, J. D. Talley, C. Chong, and B. A. Malomed, Physica D, 216, 77 (2006).
F. Smektala, C. Quemard, V. Couderc, and A. Barthélémy, J. Non-Cryst. Solids, 274, 232 (2000).
G. Boudebs, S. Cherukulappurath, H. Leblond, et al., Opt. Commun., 219, 427 (2003).
A. Zakery and S. R. Elliott, Optical Nonlinearities in Chalcogenide Glasses and Their Applications, Springer (2007).
C. Chong and D. E. Pelinovsky, Discr. Contin. Dyn. Syst. S, 4, 1019 (2011).
V. Arnold and Yu. Il’yashenko, Ordinary Differential Equations, Dynamical Systems 1, Springer-Verlag, New York (1988).
V. G. Gelfreich, V. F. Lazutkin, and M. B. Tabanov, Chaos, 1, 137 (1991).
V. G. Gelfreich and V. F. Lazutkin, Russ. Math. Surv., 56, 499 (2001).
V. Gelfreich and C. Simó, Discr. Contin. Dyn. Syst. B, 10, 511 (2008).
A. V. Savin, Y. Zolotaryuk, and J. C. Eilbeck, Physica D, 138, 267 (2000).
Ya. Zolotaryuk and I. O. Starodub, Phys. Rev. E, 91, 013202 (2015).
Yu. M. Aliev and V. P. Silin, J. Exp. Theor. Phys., 77, 142 (1993).
A. S. Malishevskii, V. P. Silin, S. A. Uryupin, and S. G. Uspenskii, Phys. Lett. A, 372, 4109 (2008).
A. S. Malishevski, V. P. Silin, and S. A. Uryupin, J. Exp. Theor. Phys., 90, 671 (2000).
G. L. Alfimov, A. S. Malishevskii, and E. V. Medvedeva, Physica D, 282, 16 (2014).
J. Yang, B. A. Malomed, and D. J. Kaup, Phys. Rev. Lett., 83, 1958 (1999).
A. R. Champneys, B. A. Malomed, J. Yang, and D. J. Kaup, Physica D, 152, 153, 340 (2001).
J. Yang, B. A. Malomed, D. J. Kaup, and A. R. Champneys, Math. Comput. Simul., 56, 585 (2001).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Alfimov, G.L., Titov, R.R. Asymptotic Formula for “Transparent Points” for Cubic–Quintic Discrete NLS Equation. J Russ Laser Res 40, 452–466 (2019). https://doi.org/10.1007/s10946-019-09826-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10946-019-09826-z