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Characterization of leapfrogging solitary waves in coupled nonlinear transmission lines

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Abstract

Leapfrogging solitary waves are characterized in two capacitively coupled transmission lines that are periodically loaded with Schottky varactors, called coupled nonlinear transmission lines (NLTLs). The coupling implies that a nonlinear solitary wave moving on one of the lines is bounded with the wave moving on the other line, which results in the periodic amplitude/phase oscillation called leapfrogging. In this study, we clarify how the leapfrogging frequency depends on the physical parameters of coupled NLTLs using a numerical model validated through measuring test lines and demonstrate the relaxation of leapfrogging. In addition, coupled Korteweg-de Vries equations are derived by applying the reductive perturbation method to the transmission equations of coupled NLTLs. Using perturbation theory based on the inverse scattering transform, a closed-form expression of leapfrogging frequency is obtained and the parameter values that simulate the properties well are examined. Engineering applications based on leapfrogging are finally discussed.

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References

  1. Gottwald, G., Grimshaw, R., Malomed, B.: Parametric envelope solitons in coupled Korteweg-de Vries equations. Phys. Lett. A 227, 47–54 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Grimshaw, R., Malomed, B.: New type of gap soliton in a coupled Korteweg-de Vries wave system. Phys. Rev. Lett. 72, 949–953 (1994)

    Article  MathSciNet  Google Scholar 

  3. Zhou, Y., Wang, M., Wang, Y.: Periodic wave solutions to a coupled KdV equations with variable coefficients. Phys. Lett. A 308, 31–36 (2003)

  4. Lou, S.Y., Tong, B., Hu, H.-C., Tang, X.-Y.: Coupled KdV equations derived from two-layer fluids. J. Phys. A Math. Gen. 39, 513–527 (2006)

    Article  MathSciNet  Google Scholar 

  5. Triki, H., El Akrmi, A., Rabia, M.K.: Soliton solutions in three linearly coupled Korteweg-de Vries equations. Opt. Commun. 201, 447–455 (2002)

    Article  Google Scholar 

  6. El-Dib, Y.O.: Nonlinear wave–wave interaction and stability criterion for parametrically coupled nonlinear Schrödinger equations. Nonlinear Dyn. 24, 399–418 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Guo, R., Liu, Y. -F., Hao, H. -Q., Qi, F. -H.: Coherently coupled solitons, breathers and rogue waves for polarized optical waves in an isotropic medium. Nonlinear Dyn. doi:10.1007/s11071-015-1938-z

  8. Liu, A.K., Kubota, T., Ko, D.R.S.: Resonant transfer of energy between nonlinear waves in neighboring pycnoclines. Stud. Appl. Math. 63, 26–46 (1980)

    MathSciNet  Google Scholar 

  9. Liu, A.K., Pereira, N.R., Ko, D.R.S.: Weakly interacting internal solitary waves in neighboring pycnoclines. J. Fluid Mech. 122, 187–194 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  10. Weidman, P.D., Johnson, M.: Experiments on leapfrogging internal solitary waves. J. Fluid Mech. 122, 195–213 (1982)

    Article  Google Scholar 

  11. Gear, J.A., Grimshaw, R.: Weak and strong interactions between internal solitary waves. Stud. Appl. Math. 70, 235–258 (1984)

    MathSciNet  MATH  Google Scholar 

  12. Nitsche, M., Weidman, P.D., Grimshaw, R., Ghrist, M., Fornberg, B.: Evolution of solitary waves in a two-pycnocline system. J. Fluid Mech. 642, 235–277 (2010)

  13. Rodwell, M.J.W., Allen, S.T., Yu, R.Y., Case, M.G., Bhattacharya, U., Reddy, M., Carman, E., Kamegawa, M., Konishi, Y., Pusl, J., Pullela, R.: Active and nonlinear wave propagation devices in ultrafast electronics and optoelectronics. Proc. IEEE 82, 1037–1059 (1994)

    Article  Google Scholar 

  14. Kintis, M., Lan, X., Fong, F.: An MMIC pulse generator using dual nonlinear transmission lines. IEEE Microw. Wirel. Compon. Lett. 17(6), 454–456 (2007)

    Article  Google Scholar 

  15. Yildirim, O.O., Ricketts, D.S., Ham, D.: Reflection soliton oscillator. IEEE Trans. Microw. Theory Tech. 57(10), 2344–2353 (2009)

    Article  Google Scholar 

  16. Malomed, B.A.: Leapfrogging solitons in a system of coupled KdV equations. Wave Motion 9, 401–411 (1987)

    Article  MathSciNet  Google Scholar 

  17. Kivshar, Y.S., Malomed, B.A.: Solitons in a system of coupled KdV equations. Wave Motion 11, 261–269 (1989)

    Article  MathSciNet  Google Scholar 

  18. Espinosa-Cerón, A., Malomed, B.A., Fujioka, J., Rodriguez, R.F.: Symmetry breaking in linearly coupled Korteweg-de Vries systems. Chaos 22, 033145-1–033145-10 (2012)

  19. Narahara, K.: Characterization of nonlinear transmission lines for short pulse amplification. J. Infrared Millim. Terahertz Waves 31, 411–421 (2009)

    MATH  Google Scholar 

  20. Wright, J.D., Scheel, A.: Solitary waves and their linear stability in weakly coupled KdV equations. Z. angew. Math. Phys. 58, 535–570 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jeffrey, A., Kawahara, T.: Asymptotic Methods in Nonlinear Wave Theory. Pitman, London (1982)

    MATH  Google Scholar 

  22. Kivshar, Y.S., Malomed, B.A.: Dynamics of solitons in nearly integrable systems. Rev. Mod. Phys. 61, 763–915 (1989)

    Article  Google Scholar 

Download references

Acknowledgments

This work was partially supported by JSPS KAKENHI Grant Number 26420296.

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Authors and Affiliations

  1. Department of Electrical and Electronic Engineering, Kanagawa Institute of Technology, 1030 Shimoogino, Atsugi, Kanagawa, 243-0292, Japan

    Koichi Narahara

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  1. Koichi Narahara
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Correspondence to Koichi Narahara.

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Narahara, K. Characterization of leapfrogging solitary waves in coupled nonlinear transmission lines. Nonlinear Dyn 81, 1805–1814 (2015). https://doi.org/10.1007/s11071-015-2108-z

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  • DOI: https://doi.org/10.1007/s11071-015-2108-z

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