Skip to main content
SpringerLink
Log in

Non Classical Solitary Waves

  • Chapter
Differential Equations Theory, Numerics and Applications

  • 246 Accesses

Abstract

This lecture will survey some recent results, mainly due to Anne de Bouard and the author, on solitary waves solutions to nonlinear dispersive equations with weak transverse effects. Typical examples are the generalized Kadomtsev Petviashvili equation or the equation for Langmuir waves in a weakly magnetized plasma. Those solitary waves are not “classical” in the sense that (contrarily to the solitary waves of the Korteweg de Vries or the usual nonlinear Schrödinger equations), they are not radial and do not decay rapidly at infinity. This is due to the breaking of radial symmetry which leads to the anisotropy of the underlying partial differential equation.

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. J. Ablowitz and J. Satsuma. Solitons and rational solutions of nonlinear evolution equations. J. Math. Phys., 19(10):2180–2186, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Albert, J. L. Bona, and D. Henry. Sufficient conditions for stability of solitary wave solutions of model equations of long waves. Physica D, 24:343–366, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  3. T. B. Benjamin. The stability of solitary waves. Proc. Roy. Soc., A328:153–183, 1972.

    MathSciNet  Google Scholar 

  4. O. V. Besov, V. P. Il’in, and S. M. Nikolskii. Integral representation of functions and imbeddings theorems, volume I. J. Wiley, 1978.

    Google Scholar 

  5. J. L. Bona. On the stability of solitary waves. Proc. Roy. Soc., A344:363–374, 1975.

    MathSciNet  Google Scholar 

  6. J. L. Bona, Souganidis P. E., and Strauss W. A. Stability and instability of solitary waves. Proc. Roy. Soc., A411:395–417, 1987.

    MathSciNet  Google Scholar 

  7. J. L. Bona and A. Li Yi. Decay and analyticity of solitary waves. Preprint, 1995.

    Google Scholar 

  8. M. J. Boussinesq. Essai sur la théorie des eaux courantes. In Mémoires présentés par divers savants à l’Académie des Sciences. Inst. France, volume 23 of 2, pages 1–680, 1877.

    Google Scholar 

  9. T. Cazenave. An introduction to nonlinear Schrödinger equations. I.M.U.F.R.J., 1992.

    Google Scholar 

  10. T. Cazenave and P. L. Lions. Orbital stability of standing waves for some nonlinear Schrödinger equations. Cornm. Math. Phys., 85:549–561, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. de Bouard and J. C. Saut. Langmuir solitary waves in a weakly magnetized plasma. In Proceedings of the Fourth MSJ International Research Institute, Sapporo, 1995. To appear.

    Google Scholar 

  12. A. de Bouard and J. C. Saut. Sur les ondes solitaires des équations de Kadomtsev-Petviashvili. C. R. Acad. Sci. Paris, 320:315–320, 1995.

    MATH  Google Scholar 

  13. A. de Bouard and J. C. Saut. Remarks on the stability of generalized KP solitary waves. In Mathematical Problems in the Theory of Water Waves, pages 75–84. Contemporary Mathematics 200 AMS, 1996.

    Google Scholar 

  14. A. de Bouard and J. C. Saut. Solitary waves of the generalized Kadomtsev-Petviashvili equations. Annales IHP, Analyse Non lineaire, 1997.

    Google Scholar 

  15. A. de Bouard and J. C. Saut. Symmetries and decay of the generalized KP solitary waves. SIAM J. Math. Analysis, 1997.

    Google Scholar 

  16. K. A. Gorshkov and D. E. Pelinovsky. Asymptotic theory of plane soliton self focusing in two dimensional wave media. Physica D, 85:468–484, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  17. F. Hamidouche. These de Doctorat. PhD thesis, Université Paris Sud, Orsay, 1997.

    Google Scholar 

  18. M. Haragus and K. Kirchgassner. Breaking of the dimension of a steady wave: some examples. Technical report, 1995. Preprint.

    Google Scholar 

  19. M. Haragus and R. L. Pego. In preparation.

    Google Scholar 

  20. E. Infeld and A. A. Senatorski. Decay of Kadomtsev-Petviashvili solitons. Phys. Rev. Letters, 72(9):1345–1347, 1994.

    Article  Google Scholar 

  21. V. Isakov. Carleman type estimates in an anisotropic case and applications. J. Diff. Eq., 105:217–238, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  22. P. Isaza, J. Mejia, and V. Stallbohm. Local solution for the Kadomtsev-Petviashvili equation in IR2. J. Math. Anal, and Appl., 196:566–587, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  23. B. B. Kadomtsev and V. I. Petviashvili. On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dok., 15:539–541, 1970.

    MATH  Google Scholar 

  24. D. J. Korteweg and G. de Vries. On the change of form of long waves advancing in a rectangular channel and on a new type of solitary waves. Phil. Mag., 39:422–443, 1895.

    MATH  Google Scholar 

  25. E. A. Kuznezov and M. M. Skoric. Hierarchy of collapse regimes for upper hybrid and lower hybrid waves. Phys. Rev. A., 38(3):1422–1426, 1988.

    Google Scholar 

  26. E. A. Kuznezov and S. K. Turitsyn. Two and Three dimensional solitons in weakly dispersive media. Sov. Phys. JETP, 55(5):844–847, 1982.

    Google Scholar 

  27. Y. Liu and X. P. Wang. Nonlinear stability of solitary waves of a generalized Kadomtsev-Petviashvili equation. Preprint, 1996.

    Google Scholar 

  28. P.I. Lizorkin. Multipliers of Fourier integrals. Proc. Steklov Inst. Math., 89:269–290, 1967.

    MATH  Google Scholar 

  29. O. Lopes. A constrained minimization problem with integrals on the entire space. J. Diff. Eq., 124:378–388, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  30. L. Molinet. Thése de Doctorat. PhD thesis, Université Paris Sud, Orsay, 1996.

    Google Scholar 

  31. Y. Murakami and M. Tajiri. Interaction between two y-periodic solitons solutions the Kadomtsev Petviashvili with positive dispersion. Wave Motion, 14:169–185, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  32. R. L. Pego and M. I. Weinstein. Strong spectral stability of solitary waves for Boussinesq equations. Preprint, 1995.

    Google Scholar 

  33. V. I. Petviashvili and V. V. Yanikov. Solitons and turbulence. Review of Plasma Physics XIV, pages 1–61, 1989.

    Google Scholar 

  34. J. C. Saut. Remarks on the generalized Kadomtsev Petviashvili equations. Indiana Math. J. 42, 3:1011–1026, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. C. Saut. Recent results on the generalized Kadomtsev-Petviashvili. Acta Applicandae Mathematicae, 39:477–487, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  36. T. Taniuti and A. Hasegawa. Wave Motion 13. 1991.

    Google Scholar 

  37. M. Tom. On a generalized Kadomtsev-Petviashvili equation. In Mathematical Problems in the Theory of Water Waves, pages 193–210. Contemporary Mathematics 200 AMS, 1996.

    Google Scholar 

  38. S. K. Turitsyn and Fal’Kovich G. E. Stability of magnetoelastic solitons and self focusing of sound in antiferromagnets. Sov. Phys. JETP 62, 1:146–152, 1985.

    Google Scholar 

  39. S. Ukai. Local solutions of the Kadomtsev-Petviashvili equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math, 36:199–209, 1989.

    MathSciNet  Google Scholar 

  40. X. P. Wang, M. J. Ablowitz, and M. Segur. Wave collapse and unstability of solitary waves for a generalized Kadomtsev-Petviashvili. Physica D, 78:97–113, 1994.

    Article  MathSciNet  Google Scholar 

  41. M. Willem. Minimax Theorems. Birkhauser, 1996.

    Google Scholar 

  42. A. A. Zaitsev. Formation of stationary nonlinear waves by superposition of solitons. Sov. Phys. Dokl., 28(9):720–722, 1983.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Analyse Numérique et EDP CNRS et Université Paris Sud, Bat 425, 91405, Orsay, France

    Jean Claude Saut

Authors
  1. Jean Claude Saut
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Applied Mathematics, University of Twente, Enschede, The Netherlands

    E. van Groesen

  2. Department of Mathematics, Institut Teknologi, Bandung, Indonesia

    E. Soewono

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Saut, J.C. (1997). Non Classical Solitary Waves. In: van Groesen, E., Soewono, E. (eds) Differential Equations Theory, Numerics and Applications. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5157-3_6

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-5157-3_6

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6168-1

  • Online ISBN: 978-94-011-5157-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation