Abstract
Nonlinearity and disorder are key players in vibrational lattice dynamics, responsible for localization and derealization phenomena. q-Breathers—periodic orbits in nonlinear lattices, exponentially localized in the reciprocal linear mode space—is a fundamental class of nonlinear oscillatory modes, currently found in disorder-free systems. In this paper we generalize the concept of q-breathers to the case of weak disorder, taking the Discrete Nonlinear Schrödinger chain as an example. We show that g-breathers retain exponential localization near the central mode, provided that disorder is sufficiently small. We analyze statistical properties of the instability threshold and uncover its sensitive dependence on a particular realization. Remarkably, the threshold can be intentionally increased or decreased by specifically arranged inhomogeneities. This effect allows us to formulate an approach to controlling the energy flow between the modes. The relevance to other model arrays and experiments with miniature mechanical lattices, light and matter waves propagation in optical potentials is discussed.
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References
E. Fermi, J. Pasta, and S. Ulam, Los Alamos Report LA-1940 (1955); Collected Papers of Enrico Fermi, Ed. by E. Segre, Vol. II (Univ. of Chicago Press, 1965), p. 977–978; Many-Body Problems, Ed. by D. C. Mattis (World Sci., Singapore, 1993).
J. Ford, Phys. Rep. 213, 271 (1992); Focus issue in Chaos 15, No. 1 (2005).
R. S. MacKay and S. Aubry, Nonlinearity 7, 1623 (1994); S. Flach and A. Gorbach, Phys. Rep. 467, 1 (2008).
P. W. Anderson, Phys. Rev. 109, 1492 (1958).
C. Albanese and J. Froehlich, Comm. Math. Phys. 138, 193 (1991); J. Archilla, R. MacKay, and J. Martin, Physica D 134, 406 (1999); G. Kopidakis and S. Aubry, Physica D 130, 155 (1999); Physica D 139, 247 (2000).
A. S. Pikovsky and D. L. Shepelyansky, Phys. Rev. Lett. 100, 094101 (2008); S. Flach, D. Krimer, and Ch. Skokos, arXiv:0805.4693 (2008); Sh. Fishman, Ye. Krivolapov, and A. Soffer, J. Stat. Phys. 131, 843 (2008).
T. Shwartz et al., Nature 446, 52 (2007).
J. Billy et al., Nature 453, 891 (2008); G. Roati et al., Nature 453, 895 (2008).
M. Sato, B. E. Habbard, and A. J. Sievers, Rev. Mod. Phys. 78, 137 (2006); M. Sato and A. J. Sievers, Low Temp. Phys. 34, 543 (2008); E. Buks and M. L. Roukes, J. Micromech. Sys. 11, 802 (2002); M. Zalalutdinov et al., Appl. Phys. Lett. 88, 143504 (2006).
Z. Y. Li et al., Nature 451, 46 (2008).
S. Flach, M. V. Ivanchenko, and O. I. Kanakov, Phys. Rev. Lett. 95, 064102 (2005); Phys. Rev. E 73, 036618 (2006).
M. V. Ivanchenko et al., Phys. Rev. Lett. 97, 025505 (2006).
K. G. Mishagin et al., New J. Phys. 10, 073034 (2008).
J. P. Nguenang, R. A. Pinto, and S. Flach, Phys. Rev. B 75, 214303 (2007).
U. Shrestha, M. Kostrun, and J. Javanainen, Phys. Rev. Lett. 101, 070406 (2008); Sh. Jia, W. Wan, and J. W. Fleischer, Opt. Lett. 32, 1668 (2007).
Yu. S. Kivshar and M. Peyrard, Phys. Rev. A 46, 3198 (1992); A. A. Morgante, M. Johansson, G. Kopidakis, and S. Aubri, Physica D 162, 53 (2002).
O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179 (2006).
Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Elsevier, Amsterdam, 2003).
J. Dieudonne, Foundations of Modern Analysis (Academic, New York, 1999).
E. W. Laedke, K. H. Spatschek, and S. K. Turitsyn, Phys. Rev. Lett. 73, 1055 (1994); E. W. Laedke et al., Phys. Rev. E 52, 5549 (1995); P. L. Christiansen et al., Phys. Scripta 67, 160 (1996).
V. E. Zakharov and A. B. Shabat Zh. Eksp. Teor. Fiz. 61, 118 (1971).
T. B. Benjamin and J. E. Feir, J. Fluid Mech. 27, 417 (1967).
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