A generalized dissipative discrete complex Ginzburg-Landau equation that governs the wave propagation in dissipative discrete nonlinear electrical transmission line with negative nonlinear resistance is derived. This equation presents arbitrarily nearest-neighbor nonlinearities. We analyze the properties of such model both in connection to their modulational stability, as well as in regard to the generation of intrinsic localized modes. We present a generalized discrete Lange-Newell criterion. Numerical simulations are performed and we show that discrete breathers are generated through modulational instability.
KeywordsGeneralized Dissipative Discrete Complex Ginzburg-Landau EquationDiscrete Lange Newell-CriterionPulse TrainsSolitary Patterns
E. Afshari and A. Hajimiri, “Nonlinear Transmission Lines for Pulse Shaping Silicon,” IEEE Journal of Solid- State Circuits, Vol. 40, No. 3, 2005, pp. 744-752. doi:10.1109/JSSC.2005.843639
E. Afshari, H. S. Bhat, A. Hajimiri and J. E. Marsden, “Extremely Wideband Signal Shaping Using One and Two Dimensional Non-Uniform Nonlinear Transmission Lines,” Journal of Applied Physics, Vol. 99, No. 5, 2006, Article ID: 054901. doi:10.1063/1.2174126
H. S. Bhat and B. Osting, “The Zone Boundary Mode in Periodic Nonlinear Electrical Lattices,” Physica D, Vol. 238, No. 14, 2009, pp. 1216-1228. doi:10.1016/j.physd.2009.04.009
H. S. Bhat and E. Afshari, “Nonlinear Constructive Interference in Electrical Lattices,” Physical Review E, Vol. 77, No. 6, 2008, Article ID: 066602. doi:10.1103/PhysRevE.77.066602
R. A. Scholtz, “Signal Selection for the Indoor Wireless Impulse Radio Channel”, IEEE 47th Vehicular Technology Conference, Vol. 3, 1997, pp. 2243-2247.
L. Q. English, S. G. Wheeler, Y. Shen, G. P. Veldes, N. Whitaker, P. G. Kevrekidis and D. J. Frantzeskakis, “Backward-Wave Propagation and Discrete Solitons in a Left-Handed Electrical Lattice,” Physics Letter A, Vol. 375, No. 9, 2011, pp. 1242-1248. doi:10.1016/j.physleta.2011.01.042
A. B. Kozyrev, H. Kim, A. Karbassi and D. W. van der Weide, “Wave Propagation in Nonlinear Left-Handed Transmission Line Media,” Applied Physics Letter, Vol. 87, No. 12, 2005, Article ID: 121109. doi:10.1063/1.2056581
A. B. Kozyrev and D. W. van der Weide, “Trains of Envelope Solitons in Nonlinear Left-Handed Transmission Line Media”, Applied Physics Letter, Vol. 91, No. 25, 2007, Article ID: 254111. doi:10.1063/1.2826282
A. B. Kozyrev and D. W. van der Weide, “Nonlinear Left-Handed Transmission Line Metamaterials” Journal of Physics D: Applied Physics, Vol. 41, No. 17, 2008, Article ID: 173001.
J. Ogasawara and K. Narahara, “Experimental Characterization of Left-Handed Transmission Lines with Regularly Spaced Schottky Varactors,” IEICE Electronics Express, Vol. 7, No. 9, 2010, p. 608. doi:10.1587/elex.7.608
Z. Wang, Y. Feng, B. Zhu, J. Zhao and T. Jiang, “Dark Schr?dinger Solitons and Harmonic Generation in Left- Handed Nonlinear Transmission Line”, Journal of Applied Physics, Vol. 107, No. 9, 2010, Article ID: 094907. doi:10.1063/1.3418556
J. M. Bilbault and P. Marquie, “Energy Localization in a Nonlinear Discrete System,” Physical Review E, Vol. 53, No. 5, 1996, pp. 5403-5408. doi:10.1103/PhysRevE.53.5403
P. Marquié, J. M. Bilbault and M. Remoissenet, “Observation of Nonlinear Localized Modes in an Electrical Lattice,” Physical Review E, Vol. 51, No. 6, 1995, pp. 6127- 6133. doi:10.1103/PhysRevE.51.6127
A. B. Togueu Motcheyo, C. Tchawoua, M. Siewe Siewe and J. D. Tchinang Tchameu, “Multisolitons and Stability of Two Hump Solitons of Upper Cutoff Mode in Discrete Electrical Transmission Line,” Physics Letter A, Vol. 375, No. 7, 2011, pp. 1104-1109. doi:10.1016/j.physleta.2011.01.018
A. Kenfack-Jiotsa and E. Talla-Tebue, “E?ect of second- neighbor inductive coupling on the modulational instability in a coupled Line of transmission,” Journal of the Physical Society of Japan, Vol. 80, 2011, Article ID: 034003. doi:10.1143/JPSJ.80.034003
D. Yemele and F. Kenmogne, “Compact Envelope Dark Solitary Wave in a Discrete Nonlinear Electrical Transmission Line” Physics Letter A, Vol. 373, No. 42, 2009, pp. 3801-3809. doi:10.1016/j.physleta.2009.08.067
E. Kengne, V. Bozicand, M. Viana and R. Vaillancourt, “Transverse Stability of Solitary Waves Propagating Incoupled Nonlinear Dispersive Transmission Lines,” Physical Review E, Vol. 78, No. 2, 2008, Article ID: 0266036. doi:10.1103/PhysRevE.78.026603
F. B. Pelap and M. M. Faye, “Higher Order Solitons in an Electrical Lattice,” Journal of the Physical Society of Japan, Vol. 76, No. 7, 2007, Article ID: 074602. doi:10.1143/JPSJ.76.074602
F. Ndzana, A. Mohamadou and T. C. Kofane, “Modulated Waves and Chaotic-Like Behaviours in the Discrete Electrical Transmission Line,” Journal of Applied Physics, Vol. 40, No. 10, 2007, pp. 3254-3262.
C. G. Lange and A. C. Newell, “A Stability Criterion for Envelope Equations” SIAM Journal of Applied Mathematics, Vol. 27, No. 3, 1974, pp. 441-456. doi:10.1137/0127034
S. Flach and A. V. Gorbach, “Discrete Breathers,” Physics Reports, Vol. 467, No. 1-3, 2008, pp. 1-116. doi:10.1016/j.physrep.2008.05.002
M. Sato, S. Yasui, M. Kimura, T. Hikihara and A. J. Sievers, “Management of Localized Energy in Discrete Nonlinear Transmission Lines” Europhysics Letters, Vol. 80, No. 3, 2007, Article ID: 30002. doi:10.1209/0295-5075/80/30002
L. Q. English, F. Palmero, A. J. Sievers, P. G. Kevrekidis and D. H. Barnak, “Traveling and Stationary Intrinsic Localized Modes and Their Spatial Control in Electrical Lattice,” Physical Review E, Vol. 81, No. 4, 2010, Article ID: 046605. doi:10.1103/PhysRevE.81.046605
H. Keubou, S. Noubissie and P. Woafo, “Waves Ampli?cation in Nonlinear Transmission Lines Using Negative Nonlinear Resistances,” Journal of the Physical Society of Japan, Vol. 73, No. 5, 2004, pp. 1147-1150. doi:10.1143/JPSJ.73.1147
J. C. Comte and P. Marquié, “Compact-Like Kink in Real Electrical Reaction—Diffusion Chain,” Chaos, Solitons & Fractals, Vol. 29, No. 2, 2006, pp. 307-312. doi:10.1016/j.chaos.2005.08.212
H. Willaime, O. Cardoso and P. Tabeling, “Frustration in a Linear Array of Vortices,” Physical Review Letters, Vol. 67, No. 23, 1991, pp. 3247-3250.
A. Mohamadou, A. K. Jiotsa and T. C. Kofane, “Modulational Instability and Unstable Patterns in the Discrete Complex Cubic Ginzburg-Landau Equation with First and Second Neighbor Couplings,” Physical Reveiw E, Vol. 72, No. 3, 2005, Article ID: 036220.
A. Minguzzi, P. Vignolo, M. L. Chiofalo and M. P. Tosi, “Hydrodynamic Excitations in a Spin-Polarized Fermi GAS under Harmonic Con?nement in One Dimension,” Physical Review A, Vol. 64, No. 3, 2001, Article ID: 033605. doi:10.1103/PhysRevA.64.033605
F. Kh. Abdullaev and M. Salerno, “Gap-Townes Solitons and Localized Excitations in Low-Dimensional BoseEinstein Condensates in Optical Lattices,” Physical Review A, Vol. 72, No. 3, 2005, Article ID: 033617. doi:10.1103/PhysRevA.72.033617
K. I. Maruno, Y. Ohta and N. Joshi, “Exact Localized Solutions of Quintic Discrete Nonlinear Schrodinger Equation” Physics Letter A, Vol. 311, No. 2, 2003, pp. 214-220. doi:10.1016/S0375-9601(03)00499-7
F. II. Ndzanan, A. Mohamadou and T. C. Kofane, “Discrete Lange-Newell Criterion for Dissipative Systems,” Physical Review E, Vol. 79, No. 5, 2009, Article ID: 056611. doi:10.1103/PhysRevE.79.056611
A. Khare, S. V. Dmitriev and A. Saxena, “Exact Moving and Stationary Solutions of a Generalized Discrete Nonlinear Schr?dinger Equation,” Journal of Physics A, Vol. 40, No. 37, 2007, pp. 11301-11317.
A. Khare, S. V. Dmitriev and A. Saxena, “Exact static Solutions of a Generalized Discrete φ4 Model Including Short-Periodic Solutions” Journal of Physics A, Vol. 42, No. 14, 2009, Article ID: 145204. doi:10.1088/1751-8113/42/14/145204
R. Stearrett and L. Q. English, “Experimental Generation of Intrinsic Localized Modes in a Discrete Electrical Transmission Line,” Journal of Physics D, Vol. 40, No. 17, 2007, pp. 5394-539. doi:10.1088/0022-3727/40/17/058
G. M. Chechin and V. P. Sakhenenko, “Interactions between Normal Modes in Nonlinear Dynamical System with Discrete Symmetry. Exact Results,” Physica D, Vol. 117, No. 1-4, 1998, pp. 43-76. doi:10.1016/S0167-2789(98)80012-2
G. M. Chechin and K. G. Zhukov, “Stability Analysis of Dynamical Regimes in Nonlinear Systems with Discrete Symmetries,” Physical Review E, Vol. 73, No. 3, 2006, Article ID: 036216.
G. M. Chechin, N. V. Novikora and A. A. Abramenko, “Bushes of Vibrational Modes for Fermi-Pasta-Ulam Chains”, Physica D, Vol. 166, No. 3-4, 2002, pp. 208-238. doi:10.1016/S0167-2789(02)00430-X