Skip to main content
SpringerLink
Log in

Soliton Ratchets in sine-Gordon-Like Equations

  • Chapter
  • First Online:
The sine-Gordon Model and its Applications

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 10))

  • 1715 Accesses

Abstract

This chapter is an overview of the soliton ratchets (motion of solitons under zero-average external forces) in the driven and damped sine-Gordon equation. Soliton ratchets induced by the breaking of spatial, temporal and field symmetries are discussed in detail. Analytical methods, such as collective coordinates theory and symmetry analysis, are presented, together with their comparison with numerical simulations. Symmetry analysis, based on the time-shift invariance of the average velocity of solitons, explains in an unified way, intrinsic phenomena of the ratchets such as: the control of reversal currents by means of the amplitudes and phases of the drivers, the motion induced by damping, and the suppression of the current when certain symmetries of the forces hold. Finally, it is pointed out that some of the previous results require verification through numerical simulations and experiments.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Along this work we will use solitons instead of solitary waves.

  2. 2.

    Solitons can be classified into topological and non-topological according to the value of the topological charge.

  3. 3.

    A functional is a mapping from a set of functions to the real numbers.

  4. 4.

    v must be n times Fréchet differentiable.

References

  1. P. Reimann, Phys. Rep. 361, 57 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. P. Hänggi, F. Marchesoni, Rev. Mod. Phys. 81, 387 (2009)

    Article  ADS  Google Scholar 

  3. S. Flach, O. Yevtushenko, Y. Zolotaryuk, Phys. Rev. Lett. 84, 2358 (2000)

    Article  ADS  Google Scholar 

  4. P. Reimann, Phys. Rev. Lett. 86, 4992 (2001)

    Article  ADS  Google Scholar 

  5. M. Salerno, Y. Zolotaryuk, Phys. Rev. E 65, 056603 (2002)

    Article  ADS  Google Scholar 

  6. F. Falo, P.J. Martínez, J.J. Mazo, T.P. Orlando, K. Segall, E. Trías, Appl. Phys. A 75, 263 (2002)

    Article  ADS  Google Scholar 

  7. A.V. Ustinov, C. Coqui, A. Kemp, Y. Zolotaryuk, M. Salerno, Phys. Rev. Lett. 93, 087001 (2004)

    Article  ADS  Google Scholar 

  8. M. Beck, E. Goldobin, M. Neuhaus, M. Siegel, R. Kleiner, D. Koelle, Phys. Rev. Lett. 95, 090603 (2005)

    Article  ADS  Google Scholar 

  9. S. Ooi, S. Savel’ev, M.B. Gaifullin, T. Mochiku, K. Hirata, F. Nori, Phys. Rev. Lett. 99, 207003 (2007)

    Google Scholar 

  10. T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina, M. Weitz, Science 326, 2141 (2009)

    Article  Google Scholar 

  11. R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. 1 (Addison-Wesley, Reading, 1963), pp. 46.1–46.9

    Google Scholar 

  12. M.O. Magnasco, Phys. Rev. Lett. 71, 1477 (1993)

    Article  ADS  Google Scholar 

  13. P. Hänggi, F. Marchesoni, F. Nori, Ann. Phys. (Leipzig) 14, 51 (2005)

    Article  MATH  ADS  Google Scholar 

  14. E.R. Kay, D.A. Leigh, F. Zerbetto, Angew. Chem. Int. 46, 72 (2007)

    Article  Google Scholar 

  15. H. Linke, B.J. Alemen, L.D. Melling, M.J. Taormina, M.J. Francis, C.C. Dow-Hygelund, V. Narayanan, R.P. Taylor, A. Stout, Phys. Rev. Lett. 96, 154502 (2006)

    Article  ADS  Google Scholar 

  16. G. Dupeux, M. Le Merrer, Ch. Clanet, D. Quéré, Phys. Rev. Lett. 107, 114503 (2011)

    Article  ADS  Google Scholar 

  17. J.H.E. Cartwright, A.G. Checa, M. Rousseau, Langmuir 29, 8370 (2013)

    Article  Google Scholar 

  18. A. Ajdari, D. Mukamel, L. Peliti, J. Prost, J. Phys. I France 4, 1551 (1994)

    Article  Google Scholar 

  19. D.R. Chialvo, M.M. Millonas, Phys. Rev. A 209, 26 (1995)

    Google Scholar 

  20. D. Cole, S. Bending, S. Savel’ev, A. Grigorenko, T. Tamegai, F. Nori, Nat. Mater. 5, 305 (2006)

    Google Scholar 

  21. W. Schneider, K. Seeger, Appl. Phys. Lett. 8, 133 (1966)

    Article  ADS  Google Scholar 

  22. R. Gommers, S. Bergamini, F. Renzoni, Phys. Rev. Lett. 95, 073003 (2005)

    Article  ADS  Google Scholar 

  23. A. Sánchez, A.R. Bishop, SIAM Rev. 40, 579 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. M. Salerno, N.R. Quintero, Phys. Rev. E. 65, 0625602 (2002)

    MathSciNet  ADS  Google Scholar 

  25. L. Morales-Molina, N.R. Quintero, F.G. Mertens, A. Sánchez, Phys. Rev. Lett. 91, 234102 (2003)

    Article  ADS  Google Scholar 

  26. L. Morales-Molina, F.G. Mertens, A. Sánchez, Eur. Phys. J. B 37, 79 (2004)

    Article  ADS  Google Scholar 

  27. Y. Zolotaryuk, M. Salerno, Phys. Rev. E. 73, 066621 (2006)

    Article  ADS  Google Scholar 

  28. J. Cuevas, B. Sánchez-Rey, M. Salerno, Phys. Rev. E. 82, 016604 (2010)

    Article  ADS  Google Scholar 

  29. A.V. Gorbach, S. Denisov, S. Flach, Optics Lett. 31, 1702 (2006)

    Article  ADS  Google Scholar 

  30. D. Poletti, E.A. Ostrovskaya, T.J. Alexander, B. Li, Yu.S. Kivshar, Physica D 238, 1338 (2009)

    Article  MATH  ADS  Google Scholar 

  31. F.G. Mertens, N.R. Quintero, A.R. Bishop, Phys. Rev. E. 81, 016608 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  32. F.G. Mertens, N.R. Quintero, I. Barashenkov, A.R. Bishop, Phys. Rev. E. 84, 026614 (2011)

    Article  ADS  Google Scholar 

  33. E. Zamora-Sillero, N.R. Quintero, F.G. Mertens, Phys. Rev. E 74, 046607 (2006)

    Article  ADS  Google Scholar 

  34. F.G. Mertens, L. Morales-Molina, A.R. Bishop, A. Sánchez,, P. Müller, Phys. Rev. E 74, 066602 (2006)

    Article  ADS  Google Scholar 

  35. V. Stehr, P. Müller, F.G. Mertens, A.R. Bishop, Phys. Rev. E 79, 036601 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  36. A. Wouk, Course of Applied Functional Analysis (Wiley, New York, 1979)

    MATH  Google Scholar 

  37. N.R. Quintero, J.A. Cuesta, R. Álvarez-Nodarse, J. Phys. A Math. Theor. 44, 425205 (2011)

    Article  ADS  Google Scholar 

  38. J.A. Cuesta, N.R. Quintero, R. Álvarez-Nodarse, Phys. Rev. X 3, 041014 (2013)

    Google Scholar 

  39. L. Morales-Molina, F.G. Mertens, A. Sánchez, Phys. Rev. E 73, 046605 (2006)

    Article  ADS  Google Scholar 

  40. D.W. McLaughlin, A.C. Scott, Phys. Rev. A 18, 1652 (1978)

    Article  ADS  Google Scholar 

  41. Yu.S. Kivshar, B.A. Malomed, Rev. Mod. Phys. 61, 763 (1989)

    Article  ADS  Google Scholar 

  42. M. Aldana-González, G. Cocho, H. Larralde, G. Martínez-Mekler, J. Theor. Biol. 220, 27 (2003)

    Article  Google Scholar 

  43. L. Morales-Molina, F.G. Mertens, A. Sánchez, Phys. Rev. E 72, 016612 (2005)

    Article  ADS  Google Scholar 

  44. M. Salerno, A.C. Scott, Phys. Rev. B 26, 2474 (1982)

    Article  ADS  Google Scholar 

  45. M.J. Rice, Phys. Rev. B 28, 3587 (1983)

    Article  ADS  Google Scholar 

  46. N.R. Quintero, A. Sánchez, F.G. Mertens, Phys. Rev. E 62, 5695 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  47. D.W. McLaughlin, A.C. Scott, Phys. Rev. A 18, 1652 (1978)

    Article  ADS  Google Scholar 

  48. N.R. Quintero, P.G. Kevrekidis, Phys. Rev. E 64, 056608 (2001)

    Article  ADS  Google Scholar 

  49. T.E. Dialynas, K. Lindenberg, G.P. Tsironis, Phys. Rev. E 56, 3976 (1997)

    Article  ADS  Google Scholar 

  50. R. Bartussek, P. Hänggi, J.G. Kissner, Europhys. Lett. 28, 459 (1994)

    Article  ADS  Google Scholar 

  51. Y. Zolotaryuk, Phys. Rev. E. 86, 026604 (2012)

    Article  ADS  Google Scholar 

  52. F. Marchesoni, Phys. Rev. Lett. 77, 2364 (1996)

    Article  ADS  Google Scholar 

  53. N.R. Quintero, B. Sánchez-Rey, J. Casado-Pascual, Phys. Rev. E. 71, 058601 (2005)

    Article  ADS  Google Scholar 

  54. N.R. Quintero, B. Sánchez-Rey, M. Salerno, Phys. Rev. E. 72, 016610 (2005)

    Article  ADS  Google Scholar 

  55. N.R. Quintero, A. Sánchez, Eur. Phys. J. B 6, 133 (1998)

    Article  ADS  Google Scholar 

  56. S. Flach, Y. Zolotaryuk, A.E. Miroshnichenko, M.V. Fistul, Phys. Rev. Lett. 88, 184101 (2002)

    Article  ADS  Google Scholar 

  57. L. Morales-Molina, N.R. Quintero, F.G. Mertens, A. Sánchez, Chaos 16, 013117 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  58. N.R. Quintero, R. Álvarez-Nodarse, F.G. Mertens, Phys. Rev. E. 80 016605 (2009)

    Article  ADS  Google Scholar 

  59. D. Cubero, V. Lebedev, F. Renzoni, Phys. Rev. E 82, 041116 (2010)

    Article  ADS  Google Scholar 

  60. X. Noblin, R. Kofman, F. Celestini, Phys. Rev. Lett. 102, 194504 (2009)

    Article  ADS  Google Scholar 

  61. T. Wulf, C. Petri, B. Liebchen, P. Schmelcher, Phys. Rev. E 86, 016201 (2012)

    Article  ADS  Google Scholar 

  62. N.R. Quintero, J.A. Cuesta, R. Álvarez-Nodarse, Phys. Rev. E 81, 030102 (2010)

    Article  ADS  Google Scholar 

  63. A. Sukstanskii, K.I. Primak, Phys. Rev. Lett. 75, 3029 (1995); Yu.S. Kivshar, A. Sánchez, Phys. Rev. Lett. 77, 582 (1996)

    Google Scholar 

  64. G. Costantini, F. Marchesoni, M. Borromeo, Phys. Rev. E 65, 051103 (2002)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We thank José Cuesta, Franz G. Mertens, Renato Álvarez and Jesús Cuevas for the critical reading of this work. We acknowledge financial support through grants FIS2011-24540, from the Ministerio de Economía y Competitividad (Spain), grants FQM207, P09-FQM-4643, from Junta de Andalucía (Spain), and a grant from the Humboldt Foundation through a Research Fellowship for Experienced Researchers SPA 1146358 STP.

Author information

Authors and Affiliations

  1. Departamento de Física Aplicada I, Universidad de Sevilla. Escuela Politécnica Superior, C/ Virgen de África, 7, 41011, Sevilla, Spain

    Niurka R. Quintero

  2. Instituto de Matemáticas de la Universidad de Sevilla (IMUS)., Edificio Celestino Mutis. Avda. Reina Mercedes s/n., 41012, Sevilla, Spain

    Niurka R. Quintero

Authors
  1. Niurka R. Quintero
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Niurka R. Quintero .

Editor information

Editors and Affiliations

  1. Departamento de Físíca Aplicada I, University of Sevilla, Sevilla, Spain

    Jesús Cuevas-Maraver

  2. Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts, USA

    Panayotis G. Kevrekidis

  3. Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts, USA

    Floyd Williams

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Quintero, N.R. (2014). Soliton Ratchets in sine-Gordon-Like Equations. In: Cuevas-Maraver, J., Kevrekidis, P., Williams, F. (eds) The sine-Gordon Model and its Applications. Nonlinear Systems and Complexity, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-06722-3_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-06722-3_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-06721-6

  • Online ISBN: 978-3-319-06722-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation