×

Introduction to nonlinear discrete systems: theory and modelling. (English) Zbl 1396.39006

Summary: An analysis of discrete systems is important for understanding of various physical processes, such as excitations in crystal lattices and molecular chains, the light propagation in waveguide arrays, and the dynamics of Bose-condensate droplets. In basic physical courses, usually the linear properties of discrete systems are studied. In this paper we propose a pedagogical introduction to the theory of nonlinear distributed systems. The main ideas and methods are illustrated using a universal model for different physical applications, the discrete nonlinear Schrödinger (DNLS) equation. We consider solutions of the DNLS equation and analyse their linear stability. The notions of nonlinear plane waves, modulational instability, discrete solitons and the anti-continuum limit are introduced and thoroughly discussed. A Mathematica program is provided for better comprehension of results and further exploration. Also, a few problems, extending the topic of the paper, for independent solution are given.

MSC:

39A12 Discrete version of topics in analysis
35Q55 NLS equations (nonlinear Schrödinger equations)
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
82D25 Statistical mechanics of crystals
82D77 Quantum waveguides, quantum wires
35C08 Soliton solutions
97M50 Physics, astronomy, technology, engineering (aspects of mathematics education)
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Kivshar, Yu S.; Agrawal, G. P., Optical Solitons: From Fibers to Photonic Crystals, (2003), New York: Academic, New York
[2] Ablowitz, M. J.; Prinari, B.; Trubatch, A. D., Discrete and Continuous Nonlinear Schrödinger Systems, (2004), Cambridge: Cambridge University Press, Cambridge · Zbl 1057.35058
[3] Flach, S.; Gorbach, A. V., Discrete breathers—advances in theory and applications, Phys. Rep., 467, 1-116, (2008) · doi:10.1016/j.physrep.2008.05.002
[4] Lederer, F.; Stegeman, G. I.; Christodoulides, D. N.; Assanto, G.; Segev, M.; Silberberg, Y., Discrete solitons in optics, Phys. Rep., 463, 1-126, (2008) · doi:10.1016/j.physrep.2008.04.004
[5] Kevrekidis, P. G., The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives, (2009), Berlin: Springer, Berlin
[6] Kartashov, Y. V.; Malomed, B. A.; Torner, L., Solitons in nonlinear lattices, Rev. Mod. Phys., 83, 247, (2011) · doi:10.1103/RevModPhys.83.247
[7] Dauxois, T.; Peyrard, M.; Ruffo, S., The Fermi–Pasta–Ulam numerical experiment: history and pedagogical perspectives, Eur. J. Phys., 26, S3, (2005) · doi:10.1088/0143-0807/26/5/S01
[8] Lévesque, L., Revisiting the coupled-mass system and analogy with a simple band gap structure, Eur. J. Phys., 27, 133, (2006) · doi:10.1088/0143-0807/27/1/014
[9] Zhang, J. M.; Dong, R. X., Exact diagonalization: the Bose–Hubbard model as an example, Eur. J. Phys., 31, 591, (2010) · doi:10.1088/0143-0807/31/3/016
[10] Newman, M. E J., Resource letter CS1: complex systems, Am. J. Phys., 79, 800, (2011) · doi:10.1119/1.3590372
[11] Liang, C., An undergraduate experiment of wave motion using a coupled-pendulum chain, Am. J. Phys., 83, 389, (2015) · doi:10.1119/1.4905842
[12] Pethick, C. J.; Smith, H., Bose–Einstein Condensation in Dilute Gases, (2008), Cambridge: Cambridge University Press, Cambridge
[13] Trombettoni, A.; Smerzi, A., Discrete solitons and breathers with dilute Bose–Einstein condensates, Phys. Rev. Lett., 86, 2353, (2001) · doi:10.1103/PhysRevLett.86.2353
[14] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical Recipes in Fortran 77: The Art of Scientific Computing, (1997), Cambridge: Cambridge University Press, Cambridge
[15] Lederer, F.; Darmanyan, S.; Kobyakov, A.; Kh Abdullaev, F., Discrete solitons in nonlinear waveguide arrays, Nonlinearity and Disorder: Theory and Applications, (2001), Dordrecht: Kluwer, Dordrecht · Zbl 1044.35091
[16] Cai, D.; Bishop, A. R.; Grønbech-Jensen, N., Localized states in discrete nonlinear Schrödinger (DNLS) equations, Phys. Rev. Lett., 72, 591, (1994) · doi:10.1103/PhysRevLett.72.591
[17] Kivshar, Yu S.; Peyrard, M., Modulational instabilities in discrete lattices, Phys. Rev. A, 46, 3198, (1992) · doi:10.1103/PhysRevA.46.3198
[18] Kivshar, Yu S.; Campbell, D. K., Peierls–Nabarro potential barrier for highly localized nonlinear modes, Phys. Rev. E, 48, 3077, (1993) · doi:10.1103/PhysRevE.48.3077
[19] Tsironis, G. P.; Molina, M. I.; Hennig, D., Generalized nonlinear impurity in a linear chain, Phys. Rev. E, 50, 2365, (1994) · doi:10.1103/PhysRevE.50.2365
[20] Molina, M. I.; Vicencio, R. A.; Kivshar, Yu S., Discrete solitons and nonlinear surface modes in semi-infinite waveguide arrays, Opt. Lett., 31, 1693, (2006) · doi:10.1364/OL.31.001693
[21] Carretero-González, R.; Talley, J. D.; Chong, C.; Malomed, B. A., Multistable solitons in the cubic-quintic discrete nonlinear Schrödinger equation, Physica D, 216, 77, (2006) · Zbl 1101.37049 · doi:10.1016/j.physd.2006.01.022
[22] Abdullaev, F. Kh; Bouketir, A.; Messikh, A.; Umarov, B. A., Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Shrödinger equation, Physica D, 232, 54, (2007) · Zbl 1125.37052 · doi:10.1016/j.physd.2007.05.005
[23] Salerno, M., Quantum deformations of the discrete nonlinear Schrödinger equation, Phys. Rev. A, 46, 6856, (1992) · doi:10.1103/PhysRevA.46.6856
[24] Gómez-Gardeñes, J.; Malomed, B. A.; Floria, L. M.; Bishop, A. R., Solitons in the Salerno model with competing nonlinearities, Phys. Rev. E, 73, (2006) · doi:10.1103/PhysRevE.73.036608
[25] Kevrekidis, P. G.; Kivshar, Yu S.; Kovalev, A. S., Instabilities and bifurcations of nonlinear impurity modes, Phys. Rev. E, 67, (2003) · doi:10.1103/PhysRevE.67.046604
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.