volume | PIER Journals

Abstract

A nonlinear electrical transmission line with an intersite circuit element acting as a nonlinear resistance is introduced and investigated. In the continuum limit, the dynamics of localized signals is described by a non-linear evolution equation belonging to the family of nonlinear diffusive Burgers' equations. This equation admits compact pulse solutions and shares some symmetry properties with the Rosenau-Hyman K(2,2) equation. An exact discrete compactly-supported signal voltage is found for the network and the dissipative effects on the pulse motion analytically studied. Numerical simulations confirm the validity of analytical results and the robustness of these compact pulse signals which may have important applications in signal processing systems.

1. Rosenau, P. and J. M. Hyman, "Compactons: Solitons with finites wavelength," Phys. Rev. Lett., Vol. 70, 564-567, 1993.
doi:10.1103/PhysRevLett.70.564 Google Scholar

2. Remoissenet, M., Waves Called Solitons, 3rd Ed., Springer-Verlag, 1999.
doi:10.1007/978-3-662-03790-4

3. Rosenau, P. and E. Kashdan, "Compactification of nonlinear patterns and waves," Phys. Rev. Lett., Vol. 101, 264101-264105, 2008.
doi:10.1103/PhysRevLett.101.264101 Google Scholar

4. Destrade, M., G. Gaeta, and G. Saccomandi, "Weierstrasss criterion and compact solitary waves," Phys. Rev. E, Vol. 75, 047601-047605, 2007.
doi:10.1103/PhysRevE.75.047601 Google Scholar

5. Gaeta, G., T. Gramchev, and S. Walcher, "Compact solitary waves in linearly elastic chains with non-smooth on-site potential," J. Phys. A: Math. Theor., Vol. 40, 4493-4509, 2007.
doi:10.1088/1751-8113/40/17/007 Google Scholar

6. Rosenau, P., "On compactification of patterns by a singular convection or stress," Phys. Rev. Lett., Vol. 99, 234102-234107, 2007.
doi:10.1103/PhysRevLett.99.234102 Google Scholar

7. Kivshar, Y. S., "Intrinsic localized modes as solitons with a compact support," Phys. Rev. E , Vol. 48, 43-45, 1993.
doi:10.1103/PhysRevE.48.R43 Google Scholar

8. Kevrekidis, P. G., V. V. Konotop, A. R. Bishop, and S. Takeno, "Discrete compactons: Some exact resuts," J. Phys. A: Math. Gen. , Vol. 35, 641-652, 2002.
doi:10.1088/0305-4470/35/45/103 Google Scholar

9. Dusuel, S., P. Michaux, and M. Remoissenet, "From kinks to compacton like kinks," Phys. Rev. E, Vol. 57, 2320-2326, 1998.
doi:10.1103/PhysRevE.57.2320 Google Scholar

10. Ludu, A. and J. P. Draayer, "Patterns on liquid surfaces cnoidal waves, compactons and scaling," Physica D, Vol. 123, 82-91, 1998.
doi:10.1016/S0167-2789(98)00113-4 Google Scholar

11. Grimshaw, R. H. J., L. A. Ostrovsky, V. I. Shrira, and Y. A. Stepanyants, "Long nonlinear surface and internal gravity waves in a rotating ocean ," Surv. Geophys. , Vol. 19, 289-338, 1998.
doi:10.1023/A:1006587919935 Google Scholar

12. Takeno, S., "Compacton-like modes in model DNA systems and their bearing on biological functioning," Phys. Lett. A, Vol. 339, 352-360, 2005.
doi:10.1016/j.physleta.2005.01.081 Google Scholar

13. Rosenau, P. and A. Pikovsky, "Phase compactons in chains of dispersively coupled oscillators," Phys. Rev. Lett., Vol. 94, 174102-174106, 2005.
doi:10.1103/PhysRevLett.94.174102 Google Scholar

14. Pikovsky, A. and P. Rosenau, "Phase compactons," Physica D, Vol. 218, 56-69, 2006.
doi:10.1016/j.physd.2006.04.015 Google Scholar

15. Takahashi, D. and J. Satsuma, "Explicit solutions of magma equation," J. Phys. Soc. Jpn., Vol. 57, 417-421, 1988.
doi:10.1143/JPSJ.57.417 Google Scholar

16. Simpson, G., M. I. Weinstein, and P. Rosenau, "On a hamiltonian PDE arising in magma dynamics," Disc. and Cont. Dynamical Systems B, Vol. 10, 903-924, 2008.
doi:10.3934/dcdsb.2008.10.903 Google Scholar

17. Gharakhili, F. G., M. Shahabadi, and M. Hakkak, "Bright and dark soliton generation in a left-handed nonlinear transmission line with series ," Progress In Electromagnetics Research, Vol. 96, 237-249, 2009.
doi:10.2528/PIER09080106 Google Scholar

18. Afshari, E., H. S. Bhat, A. Hajimiri, and J. E. Marsden, "Extremely wideband signal shaping using one and two dimensional nonuniform nonlinear line ," J. Appl. Phys., Vol. 99, 054901-054917, 2006.
doi:10.1063/1.2174126 Google Scholar

19. Narahara, , K. and M. Nakamura, "Compensation of polarization mode dispersion with electrical nonlinear transmission lines," Jpn. J. Appl. Phys., Vol. 42, 6327-6334, 2003.
doi:10.1143/JJAP.42.6327 Google Scholar

20. Narahara, K., "Coupled nonlinear transmission lines for doubling repetition rate of incident pulse streams," Progress In Electromagnetics Research Letters , Vol. 16, 69-78, 2010.
doi:10.2528/PIERL10070106 Google Scholar

21. Narahara, K., "Characterization of partially nonlinear transmission lines for ultrashort-pulse amplification," Jpn. J. Appl. Phys., Vol. 42, 5508-5515, 2003.
doi:10.1143/JJAP.42.5508 Google Scholar

22. Comte, J. C. and P. Marquie, "Compact-like kink in real electrical eaction-diffusion chain," Chaos, Soliton, Fractals, Vol. 29, 307-312, 2006.
doi:10.1016/j.chaos.2005.08.212 Google Scholar

23. Yemele, D. and F. Kenmogne, "Compact envelope dark solitary wave in a discrete nonlinear electrical transmission line," Phys. Lett. A, Vol. 373, 3801-3809, 2009.
doi:10.1016/j.physleta.2009.08.067 Google Scholar

24. Kenmogne, F. and D. Yemele, "Exotic modulated signals in a nonlinear electrical transmission line: Modulated peak solitary wave and gray compacton," Chaos, Solitons, Fractals, Vol. 45, 21-34, 2012.
doi:10.1016/j.chaos.2011.09.009 Google Scholar

25. English, L. Q., R. Basu Thakur, and R. Stearrett, "Patterns of travelling intrinsic localized modes in a driven electrical lattice," Phys. Rev. E, Vol. 77, 066601-066605, 2008.
doi:10.1103/PhysRevE.77.066601 Google Scholar

26. Marquie, P., S. Binczak, J. C. Comte, B. Michaux, and J. M. Bilbault, "Diffusion effects in a nonlinear electrical lattice," Phys. Rev. E, Vol. 57, 6075-6078, 1998.
doi:10.1103/PhysRevE.57.6075 Google Scholar

27. Comte, J. C., P. Marquie, J. M. Bilbault, and S. Binczak, "Noise removal using a nonlinear two-dimensional diffusion network," Ann. Telecommun., Vol. 53, 483-487, 1998. Google Scholar

28. Nguena, H. K., S. Noubissi, and P.Woafo, "Waves amplification in nonlinear transmission lines using negative nonlinear resistance," J. Phys. Soc. Jpn., Vol. 73, 1147-1150, 2004.
doi:10.1143/JPSJ.73.1147 Google Scholar

29. Ndzana, F., A. Mohamadou, and T. C. Kofane, "Modulated waves and chaotic-like behaviours in the discrete electrical line," J. Phys. D: Appl. Phys., Vol. 40, 3254-3262, 2007.
doi:10.1088/0022-3727/40/10/035 Google Scholar

30. Binzak, S., J. C. Comte, B. Michaux, P. Marquie, and and, "Experimental nonlinear electrical reactiondiffusion lattice," Electron. Lett., Vol. 34, 1061-1062, 1998.
doi:10.1049/el:19980774 Google Scholar

31. Saccomandi, G. and I. Sgura, "The relevance of nonlinear stacking interactions in simple models of double-stranded DNA," J. R. Soc. Interface, Vol. 3, 655-667, 2006.
doi:10.1098/rsif.2006.0126 Google Scholar

32. Nguetcho, A. S., J. R. Bogning, D. Yemele, and T. C. Kofane, "Kink compactons in models with parametrized periodic double-well and asymmetric substrate potentials," Chaos, Solitons Fractals, Vol. 21, 165-176, 2004.
doi:10.1016/j.chaos.2003.10.034 Google Scholar

Dr. Desire Ndjanfang

Dr. David Yemele

Dr. Patrick Marquie

Dr. Timoleon Crepin Kofane