Modulated waves and pattern formation in coupled discrete nonlinear LC transmission lines.

The conditions for the propagation of modulated waves on a system of two coupled discrete nonlinear LC transmission lines with negative nonlinear resistance are examined, each line of the network containing a finite number of cells. Our theoretical analysis shows that the telegrapher equations of the electrical transmission line can be reduced to a set of two coupled discrete complex Ginzburg-Landau equations. Using the standard linear stability analysis, we derive the expression for the growth rate of instability as a function of the wave numbers and system parameters, then obtain regions of modulational instability. Using numerical simulations, we examine the long-time dynamics of modulated waves in the line. This leads to the generation of nonlinear modulated waves which have the shape of a soliton for the fast and low modes. The effects of dissipative elements on the propagation are also shown. The analytical results are corroborated by numerical simulations.

[1]  W. M. Liu,et al.  Exact solutions of the derivative nonlinear Schrödinger equation for a nonlinear transmission line. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Tetsuro Suzuki.  Non-Linear Mechanical Model for Martensitic Transformation , 1978 .

[3]  Modulational instability of partially coherent signals in electrical transmission lines. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Patrick Marquié,et al.  Noise removal using a nonlinear two-dimensional diffusion network , 1998, Ann. des Télécommunications.

[5]  Hiroyuki Nagashima,et al.  Experiment on the Toda Lattice Using Nonlinear Transmission Lines , 1978 .

[6]  T. Kofané,et al.  Wave Modulations in the Nonlinear Biinductance Transmission Line , 2001 .

[7]  Stability analysis of plane wave solutions of the discrete ginzburg-landau equation , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[8]  K. Stewartson,et al.  A non-linear instability theory for a wave system in plane Poiseuille flow , 1971, Journal of Fluid Mechanics.

[9]  Lars Q. English,et al.  Experimental generation of intrinsic localized modes in a discrete electrical transmission line , 2007, 0706.1211.

[10]  M. Ablowitz,et al.  Long-time dynamics of the modulational instability of deep water waves , 2001 .

[11]  A Smerzi,et al.  Discrete solitons and breathers with dilute Bose-Einstein condensates. , 2001, Physical review letters.

[12]  Ryogo Hirota,et al.  Studies on Lattice Solitons by Using Electrical Networks , 1970 .

[13]  A. Sievers,et al.  Modulational instability of nonlinear spin waves in easy-axis antiferromagnetic chains , 1998 .

[14]  M. Cross,et al.  Pattern formation outside of equilibrium , 1993 .

[15]  Cardoso,et al.  Frustration in a linear array of vortices. , 1991, Physical review letters.

[16]  D. Christodoulides,et al.  Discrete Ginzburg-Landau solitons. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  C. R. Willis,et al.  Discrete Breathers , 1997 .

[18]  J. Bilbault,et al.  Long-time dynamics of modulated waves in a nonlinear discrete LC transmission line. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Marquié,et al.  Modulational instability of two counterpropagating waves in an experimental transmission line. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  David C. Hutchings,et al.  Polarization stability of solitons in birefringent optical fibers , 1999 .

[21]  P. Woafo,et al.  Waves amplification in nonlinear transmission lines using negative nonlinear resistances , 2004 .

[22]  F. B. Pelap,et al.  Solitonlike excitations in a one-dimensional electrical transmission line , 2005 .

[23]  Morikazu Toda,et al.  Wave Propagation in Anharmonic Lattices , 1967 .

[24]  Exact Phase Solutions of Nonlinear Oscillators on Two-dimensional Lattice , 2003, nlin/0309009.

[25]  J. Bilbault,et al.  Bistability and nonlinear standing waves in an experimental transmission line , 1993 .

[26]  Alan J. Heeger,et al.  Solitons in polyacetylene , 1979 .

[27]  J. Bilbault,et al.  Generation of envelope and hole solitons in an experimental transmission line , 1994 .

[28]  H. Winful,et al.  Dynamics of phase-locked semiconductor laser arrays , 1988 .

[29]  W. M. Liu,et al.  Modulational instability criteria for coupled nonlinear transmission lines with dispersive elements. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  Shikuo Liu,et al.  Collision Interactions of Solitons in a Baroclinic Atmosphere , 1995 .

[31]  Kim Ø. Rasmussen,et al.  THE DISCRETE NONLINEAR SCHRÖDINGER EQUATION: A SURVEY OF RECENT RESULTS , 2001 .

[32]  T. Kofané,et al.  Localized Solitary Signals on a Coupled Nonlinear Transmission Line , 1995 .

[33]  J. Leon,et al.  Cutoff solitons and bistability of the discrete inductance-capacitance electrical line: theory and experiments. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  T. Kakutani,et al.  Solitary and Shock Waves on a Coupled Transmission Line , 1980 .

[35]  I. Aranson,et al.  The world of the complex Ginzburg-Landau equation , 2001, cond-mat/0106115.

[36]  T. Kakutani,et al.  Solitary Waves on a Two-Layer Fluid , 1978 .

[37]  A. Davydov,et al.  The theory of contraction of proteins under their excitation. , 1973, Journal of theoretical biology.

[38]  J. Bilbault,et al.  Observation of nonlinear localized modes in an electrical lattice. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[39]  T. Kofané,et al.  Suppression of the fast and slow modulated waves mixing in the coupled nonlinear discrete LC transmission lines , 2006 .

[40]  R. Hirota,et al.  Theoretical and experimental studies of lattice solitons in nonlinear lumped networks , 1973 .

[41]  T. Kofané,et al.  Gap solitons on a coupled nonlinear transmission line , 1997 .

[42]  Alidou Mohamadou,et al.  Modulated waves and chaotic-like behaviours in the discrete electrical transmission line , 2007 .

[43]  Lee A. Segel,et al.  Non-linear wave-number interaction in near-critical two-dimensional flows , 1971, Journal of Fluid Mechanics.

[44]  Patrick Marquié,et al.  DIFFUSION EFFECTS IN A NONLINEAR ELECTRICAL LATTICE , 1998 .

[45]  A. Sievers,et al.  Intrinsic localized modes in anharmonic crystals. , 1988, Physical review letters.