Skip to main content
SpringerLink
Log in

The application of Bäcklund transforms to physical problems

  • Conference paper
  • First Online:
Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 515))

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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. A.C. SCOTT, Steady propagation on long Josephson junctions, Bull. Amer. Phys. Soc. 12 (1967), 308–309.

    Google Scholar 

  2. -, A nonlinear Klein-Gordon equation, Amer. J. Phys. 37 (1969), 52–61.

    Article  Google Scholar 

  3. A.C. SCOTT AND W.J. JOHNSON, Internal flux motion in large Josephson junctions, Appl. Phys. Lett. 14 (1969), 316–318.

    Article  Google Scholar 

  4. A.C. SCOTT, Propagation of flux on a long Josephson tunnel junction, Nuovo Cimento B 69 (1970), 241–261.

    Article  Google Scholar 

  5. A. BARONE, F. ESPOSITO, C.J. MAGEE, AND A.C. SCOTT, Theory and applications of the sine-Gordon equation, Riv. Nuovo Cimento 1 (1971), 227–267.

    Article  Google Scholar 

  6. H.T. DAVIS, Introduction to Nonlinear Integral and Differential Equations, Dover Publications, New York, N. Y., 1962, Chap. 3.

    Google Scholar 

  7. D.W. MCLAUGHLIN AND A.C. SCOTT, A restricted Bäcklund transformation, J. Mathematical Phys. 14 (1973), 1817–1828.

    Article  MathSciNet  MATH  Google Scholar 

  8. P.D. GERBER, Some results on quadratic semilinear differential equations, presented at the Research Workshop on Contact Transformations.

    Google Scholar 

  9. C. ROGERS, Baecklund transformations and invariance properties in axiallysymmetric flow, Ann. Soc. Sci. Bruxelles Sér. I 86 (1972), 211–219.

    Google Scholar 

  10. D.S. MOSELEY, Nonseparable solutions of the Helmholtz wave equation, Quart. Appl. Math. 22 (1965), 354–357; Further properties of the nonseparable solutions of the Helmholtz wave equation, ibid. 27 (1969), 451–459.

    MathSciNet  MATH  Google Scholar 

  11. D.H. SCHOONAERT AND P.J. LUYPAERT, Use of nonseparable solutions of Helmholtz wave equations in waveguides and cavities, Electron. Lett. 9 (1973), 617–618.

    Article  Google Scholar 

  12. A.C. SCOTT, The electrophysics of a nerve fiber, Rev. Modern Phys. 47 (1975), 487–533.

    Article  Google Scholar 

  13. -, Analysis of nonlinear distributed systems, IRE Trans. Circuit Theory CT-9 (1962), 192–195.

    Article  Google Scholar 

  14. J. NAGUMO, S. ARIMOTO, AND S. YOSHIZAWA, Bistable transmission lines, IEEE Trans. Circuit Theory CT-12 (1965), 400–412.

    Article  Google Scholar 

  15. A.C. SCOTT, Active and Nonlinear Wave Propagation in Electronics, John Wiley and Sons, New York, N. Y., 1970.

    Google Scholar 

  16. G.B. WHITHAM, Linear and Nonlinear Waves, John Wiley and Sons, New York, N. Y., 1974.

    MATH  Google Scholar 

  17. L.S. HOEL, W.H. KELLER, J.E. NORDMAN, AND A.C. SCOTT, Niobium superconductive tunnel diode integrated circuit arrays, Solid-State Electron. 15 (1972), 1167–1173.

    Article  Google Scholar 

  18. T.A. FULTON, R.C. DYNES, AND P.W. ANDERSON, The flux shuttle—A Josephson junction shift register employing single flux quanta, Proc. IEEE 61 (1973), 28–35.

    Article  Google Scholar 

  19. J. RUBINSTEIN, Sine-Gordon Equation, J. Mathematical Phys. 11 (1970), 258–266.

    Article  MathSciNet  Google Scholar 

  20. A. SEEGER, H. DONTH, AND A. KOCHENDÖRFER, Theorie der Versetzungen in eindimensionalen Atomreihen, Zeit. f. Physik 127 (1950), 533–550; 130 (1951), 321–336; 134 (1953), 173–193.

    Article  MATH  Google Scholar 

  21. G.L. LAMB, JR., Analytical descriptions of ultrashort optical pulse propagation in a resonant medium, Rev. Modern Phys. 43 (1971), 99–124.

    Article  MathSciNet  Google Scholar 

  22. A.C. SCOTT AND F.Y.F. CHU, Pulse saturation in a traveling wave parametric amplifier, Proc. IEEE 62 (1974), 1720–1721.

    Article  Google Scholar 

  23. A.C. SCOTT, F.Y.F. CHU, AND D.W. MCLAUGHLIN, The soliton: a new concept in applied science, Proc. IEEE 61 (1973), 1443–1483.

    Article  MathSciNet  Google Scholar 

  24. A.R. FORSYTH, Theory of Differential Equations, Vol. VI, Dover Publications, New York, N. Y., 1959, p. 454.

    Google Scholar 

  25. M.J. ABLOWITZ, D.J. KAUP, A.C. NEWELL, AND H. SEGUR, The inverse scattering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math. 53 (1974), 249–315.

    Article  MathSciNet  MATH  Google Scholar 

  26. F.Y.F. CHU AND A.C. SCOTT, Bäcklund transformations and the inverse method, Phys. Lett., 47A (1974), 303–304.

    Article  MathSciNet  Google Scholar 

  27. W.J. JOHNSON, Nonlinear wave propagation on superconducting tunneling junctions, Ph.D. Thesis, University of Wisconsin, 1968.

    Google Scholar 

  28. K. NAKAJIMA, T. YAMASHITA, AND Y. ONODERA, Mechanical analog of active Josephson transmission line, J. Appl. Phys. 45 (1974), 3141–3145.

    Article  Google Scholar 

  29. K. NAKAJIMA, Y. ONODERA, T. NAKAMURA, AND R. SATO, Analysis of vortex motions on Josephson line, J. Appl. Phys. 45 (1974), 4095–4099.

    Article  Google Scholar 

  30. R. HIROTA, Exact three-soliton solution of the two-dimensional sine-Gordon equation, J. Phys. Soc. Japan 35 (1973), 1566.

    Article  Google Scholar 

  31. J. BOUSSINESQ, Theorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl. 7 (1872), 55–108.

    MathSciNet  Google Scholar 

  32. N.J. ZABUSKY, A synergetic approach to problems of nonlinear dispersive wave propagation and interaction, in Nonlinear Partial Differential Equations, W.F. Ames, Ed., Academic Press, New York, N. Y., 1967, 223–258.

    Chapter  Google Scholar 

  33. M. TODA, Studies on a nonlinear lattice, Phys. Rep. 18C (1975), 1–124.

    Article  MathSciNet  Google Scholar 

  34. R. HIROTA, Exact N-soliton solution of the wave equation of long waves in shallow-water and in nonlinear lattices, J. Mathematical Phys. 14 (1973), 810–815.

    Article  MathSciNet  MATH  Google Scholar 

  35. K.E. LONNGREN, H.C.S. HSUAN, AND W.F. AMES, On the soliton, invariant and shock solutions of a fourth-order nonlinear equation, J. Math. Anal. Appl., to be published.

    Google Scholar 

  36. N. HERSHKOWITZ AND T. ROMESSER, Observation of ion-acoustic cylindrical solitons, Phys. Rev. Lett. 32 (1974), 581–583.

    Article  Google Scholar 

  37. S. MAXON AND J. VIECELLI, Spherical solitons, Phys. Rev. Lett. 32 (1974), 4–6.

    Article  Google Scholar 

  38. V.E. ZAKHAROV, On stochastization of one-dimensional chains of nonlinear oscillators, Ž. Èksperimental’noi i Teoretičeskoî Fiziki 65 (1973), 219–225 (in Russian).

    Google Scholar 

  39. H.D. WAHLQUIST AND F.B. ESTABROOK, Bäcklund transformation for solutions of the Korteweg-deVries equation, Phys. Rev. Lett. 31 (1973), 1386–1390.

    Article  MathSciNet  Google Scholar 

  40. H.-H. CHEN, Relation between Bäcklund transformations and inverse scattering problems, this volume.

    Google Scholar 

  41. R. HIROTA, Direct method of finding exact solutions of nonlinear evolution equations, this volume.

    Google Scholar 

Download references

Author information

Author notes

  1. Alwyn C. Scott

    Present address: Mathematics Research Center, University of Wisconsin, 53706, Madison, Wisconsin

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Wisconsin, 53706, Madison, Wisconsin

    Alwyn C. Scott

Authors
  1. Alwyn C. Scott
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Robert M. Miura

Rights and permissions

Reprints and permissions

Copyright information

© 1976 Springer-Verlag

About this paper

Cite this paper

Scott, A.C. (1976). The application of Bäcklund transforms to physical problems. In: Miura, R.M. (eds) Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications. Lecture Notes in Mathematics, vol 515. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0081164

Download citation

  • DOI: https://doi.org/10.1007/BFb0081164

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-07687-2

  • Online ISBN: 978-3-540-38220-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation