Skip to main content
SpringerLink
Log in

A geometric variational formalism for the theory of nonlinear waves

  • Conference paper
  • First Online:
Geometric Methods in Mathematical Physics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 775))

  • 496 Accesses

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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.

Bibliography

  1. I.M. Krichever, "Methods of Algebraic Geometry in the Theory of Nonlinear Equations", Russian Math. Surveys 32 (1977), 125–213.

    Article  MathSciNet  MATH  Google Scholar 

  2. J.L. Burchnall and T.W. Chaundy, Proc. London Math. Soc. 21 (1922), 420–440.

    MathSciNet  Google Scholar 

  3. H. Wahlquist and F. Estabrook, "Prolongation Structures of Nonlinear Evolution Equations", J. Math. Phys. 16 (1975), 1–7.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. R. Hermann, The Geometry of Nonlinear Differential Equations, Bäcklund Transformations and Solitons, Parts A and B, Math Sci Press, Brookline, MA.

    Google Scholar 

  5. I.M. Gelfand and L.A. Dikii, "Resolvents and Hamiltonian Systems", Functional Anal. Appl. 11 (1977), 93–105.

    Article  MathSciNet  Google Scholar 

  6. Yu. Manin, "Algebraic Aspects of Nonlinear Differential Equations" (in Russian), Modern Problems in Mathematics 11 (1978), 5–152, Viniti, Moscow.

    MathSciNet  Google Scholar 

  7. Yu. Manin and B. Kupershmidt, "Long-Wave Equation with Free Boundaries", Funct. Anal. Appl. 11 (1978), 188.

    Article  MATH  Google Scholar 

  8. R. Hermann, Yang-Mills, Kaluza-Klein and the Einstein Program, Interdisciplinary Mathematics, Vol. 19, Math Sci Press, Brookline, MA. 1968.

    Google Scholar 

  9. R. Hermann, Differential Geometry and the Calculus of Variations, 2nd Edition, Interdisciplinary Mathematics, Volume 17, Math Sci Press, Brookline, MA.

    Google Scholar 

  10. R. Hermann, Lie Algebras and Quantum Mechanics, W.A. Benjamin, 1970.

    Google Scholar 

  11. R. Hermann, "The Second Variation for Variational Problems in Canonical Form", Bull. Amer. Math. Soc. 71 (1965), 145–148.

    Article  MathSciNet  MATH  Google Scholar 

  12. R. Hermann, "The Second Variation for Minimal Submanifolds", J. Math. and Mech. 16 (1966), 473–492.

    MathSciNet  MATH  Google Scholar 

  13. P. Dedecker, "On the Generalization of Symplectic Geometry to Multiple Integrals in the Calculus of Variations", in Springer Math. Lecture Notes No. 520, K. Bleuler and A. Reetz (eds.), Springer-Verlag, 1977.

    Google Scholar 

  14. W.F. Shadwick, "The Hamilton-Cartan Formalism for Higher Order Conserved Currents", Preprint, Kings College, London.

    Google Scholar 

  15. Y. Kodama and M. Wadati, "Theory of Canonical Transformations for Nonlinear Evolution Equations", I & II, Prog. Theor. Phys. 56 (1976), 1740 and 57 (1977), 1900.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. J. Moser (ed.), Lecture Notes in Physics, Vol. 38, Springer-Verlag, 1975.

    Google Scholar 

  17. R.M. Miura (ed.), Lecture Notes in Mathematics, Vol. 15, Springer-Verlag, 1976.

    Google Scholar 

  18. G. Whitham, Linear and Non-Linear Waves, Wiley, New York, 1974.

    MATH  Google Scholar 

  19. M.J. Ablowitz, D.J. Kaup, A.C. Newell, and H. Segur, Studies in Applied Math 53 (1974), 249.

    Article  MathSciNet  Google Scholar 

  20. H. Morris, "Prolongation Structures and Nonlinear Evolution Equations in Two Spatial Dimensions", J. Math. Phys. 17 (1976), 1870–1872.

    Article  ADS  MathSciNet  Google Scholar 

  21. R. Hermann, "The Pseudopotentials of Estabrook and Wahlquist, the Geometry of Solitons and the Theory of Connections", Phys. Rev. Lett. 36 (1976), 835.

    Article  ADS  MathSciNet  Google Scholar 

  22. J. Corones, "Solitons, Pseudopotentials and Certain Lie Algebras", J. Math. Phys. 18 (1977), 163–164.

    Article  ADS  MathSciNet  Google Scholar 

  23. R. Hermann, Toda Lattices, Cosymplectic Manifolds, Bäcklund Transformations and Kinks, Parts A and B, Math Sci Press, Brookline, MA.

    Google Scholar 

  24. R. Hermann, "'Modern’ Differential Geometry in Elementary Particle Physics", Proceedings of the VII GIFT Summer School, Lecture Notes in Physics, J. Azcarraga (ed.), Springer-Verlag.

    Google Scholar 

  25. R. Hermann, "The Lie-Cartan Geometric Theory of Differential Equations and Scattering Theory", to appear, Proceedings of the 1977 Park City (Utah) Conference on Differential Equations, P. Bynes (ed.)

    Google Scholar 

  26. R. Hermann, "Prolongation, Bäcklund Transformations and Lie Theory as Algorithms for Solving and Understanding Nonlinear Differential Equations", in Solitons in Action, K. Lonngren and A. Scott (eds.), Academic Press, New York, 1978.

    Google Scholar 

  27. M. Crampin, F.A.E. Pirani and D.C. Robinson, "The Soliton Connection" (to appear, Letters in Math. Phys.).

    Google Scholar 

  28. R. Hermann (ed.), The 1976 Ames Research Center (NASA) Conference on the Geometric Theory of Nonlinear Waves (articles by Estabrook, Wahlquist, Hermann, Morris, Corones, R. Gardner, and Scott), Math Sci Press, Brookline, MA.

    Google Scholar 

  29. F. Estabrook and H. Wahlquist, "Prolongation Structures, Connection Theory and Bäcklund Transformations", Proceedings of the International Symposium on Nonlinear Evolution Equations..., F. Calogero (ed.), Research Notes in Math., Pitman, London, 1978.

    Google Scholar 

  30. S. Chern and C. Terng, "Analogue of the Bäcklund Theorem for Affinely Connected Manifolds", Preprint, Math. Dept., University of California, Berkeley, 1977.

    Google Scholar 

  31. R. Hermann, Cartanian Geometry, Nonlinear Waves, and Control Theory, Part A, Interdisciplinary Mathematics, Vol. 20, Math Sci Press, Brookline, MA. 1979.

    MATH  Google Scholar 

Download references

Authors
  1. Robert Hermann
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Gerald Kaiser Jerrold E. Marsden

Rights and permissions

Reprints and permissions

Copyright information

© 1980 Springer-Verlag

About this paper

Cite this paper

Hermann, R. (1980). A geometric variational formalism for the theory of nonlinear waves. In: Kaiser, G., Marsden, J.E. (eds) Geometric Methods in Mathematical Physics. Lecture Notes in Mathematics, vol 775. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092025

Download citation

  • DOI: https://doi.org/10.1007/BFb0092025

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09742-6

  • Online ISBN: 978-3-540-38571-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation