Skip to main content
SpringerLink
Log in
Book cover

KdV ’95 pp 229–244Cite as

Integrable Nonlinear Evolution Equations and Dynamical Systems in Multidimensions

  • Chapter

Abstract

The investigation of nonlinear evolution equations and dynamical systems integrable in multidimensions constitutes at present our main research interest. Here we survey findings obtained recently as well as over time: solvable equations (both PDEs and ODEs) are reported, philosophical motivations and methodological approaches are outlined. For more detailed treatments, including the display and analysis of solutions, the interested reader is referred to the original papers.

Mathematics Subject Classifications (1991).35Q20, 58F07.

This is a preview of subscription content, 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Korteweg, D. J. and de Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag. 39 (1895), 422–443.

    MATH  Google Scholar 

  2. Gardner, C. S., Greene, J. M., Kruskal, M. D., and Miura, R. M.: Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967), 1095–1097.

    Article  MATH  Google Scholar 

  3. Zakharov, V. E. and Shabat, A. B.: Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media, Soviet Physics JETP 34 (1972), 62–69.

    MathSciNet  Google Scholar 

  4. Ablowitz, M. J., Kaup, D. J., Newell, A. C, and Segur, H.: The inverse scattering transform -Fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974), 249–315.]

    MathSciNet  Google Scholar 

  5. Zakharov, V. E., Takhtadzjan, L. A., and Faddeev, L. D.: A complete description of the solution of the sine-Gordon equation, Soviet Phys. Dokl 19 (1975), 824–826.

    Google Scholar 

  6. Calogero, F.: Why are certain nonlinear PDEs both widely applicable and integrable?, in: Zakharov, V. E. (Ed.) What is Integrabilityl, Springer, 1991, pp. 1–62.

    Chapter  Google Scholar 

  7. Lax, P. D.: Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math. 21 (1968), 467–490.

    Article  MathSciNet  MATH  Google Scholar 

  8. Toda, M.: Vibration of a chain with nonlinear interaction, J. Phys. Soc. Japan 29 (1967), 431–436

    MathSciNet  Google Scholar 

  9. Toda, M.:Waves in nonlinear lattice, Prog. Theor. Phys. Suppl. 45 (1970), 174–200.

    Article  Google Scholar 

  10. Calogero, F.: Solution of the one-dimensional AT-body problem with quadratic and/or inverselyquadratic pair potentials, J. Math. Phys. 12 (1971), 419–436.

    Article  MathSciNet  Google Scholar 

  11. Manakov, S. V: Complete integrability and stochastization of discrete dynamical systems, Sov. Phys. JETP (1974), 269–274.

    Google Scholar 

  12. Flaschka, H.: The Toda lattice. I and II, Phys. Rev. B9 (1974), 1924–1925

    MathSciNet  Google Scholar 

  13. Flaschka, H.: Prog. Theor. Phys. 51 (1974), 703–716.

    Article  MathSciNet  Google Scholar 

  14. Moser, J.: Three integrable Hamiltonian systems connected with isospectral deformations, Adv. in Math. 16 (1975), 197–220.

    Article  MATH  Google Scholar 

  15. Calogero, F.: Exactly solvable one-dimensional many-body problems, Lett. Nuovo Cimento 13 (1975), 411–416

    Article  MathSciNet  Google Scholar 

  16. Calogero, F.: Integrable dynamical systems and related mathematical results, in: Wolf, K. B. (Ed.) Nonlinear Phenomena, Lecture Notes in Physics 189, Springer, 1983, pp. 47–109;

    Chapter  Google Scholar 

  17. Calogero, F.: Integrable dynamical systems and some other mathematical results (remarkable matrices, identities, basic hypergeometric functions), in: Winternitz, P. (Ed.) Systèmes Dynamiques Non Linéaires: Integrabilité et Comportement Qualitatif, Presse de l’Université de Montréal, SMS 120, 1986, pp. 40–70

    Google Scholar 

  18. Calogero, F.: ;Remarks on certain integrable one-dimensional many-body problems, Physics Letters A183 (1993), 85–88.

    MathSciNet  Google Scholar 

  19. Calogero, F.: Olshanetsky, M. A. and Perelomov, A. M.: Classical integrable finite-dimensional systems related to Lie algebras and Quantal integrable systems related to Lie algebras, Physics Reports 71 (1981), 313–400

    Article  MathSciNet  Google Scholar 

  20. Calogero, F.: Olshanetsky, M. A. and Perelomov, A. M.: Classical integrable finite-dimensional systems related to Lie algebras and Quantal integrable systems related to Lie algebras, Physics Reports 94 (1983), 313–404.

    Article  MathSciNet  Google Scholar 

  21. Kruskal, M. D.: Lectures in App. Math. 15 (1974), 61–83.

    MathSciNet  Google Scholar 

  22. Thickstun W. R.: J. Math. Anal. Appl. 55 (1976), 335.

    Article  MathSciNet  MATH  Google Scholar 

  23. Airault, H., McKean, H. P., and Moser, J.: Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Commun. Pure Appl. Math. 30 (1977), 95–148.

    Article  MathSciNet  MATH  Google Scholar 

  24. Choodnovsky, D. V. and Choodnovsky, G. V: Pole expansions of nonlinear partial differential equations, Nuovo Cimento 40B (1977), 339–353.

    MathSciNet  Google Scholar 

  25. Calogero, F.: Motion of poles and zeros of special solutions of nonlinear and linear partial differential equations and related ‘solvable’ many-body problems, Nuovo Cimento 43B (1978), 177–241.

    MathSciNet  Google Scholar 

  26. Calogero, F.: A class of solvabe dynamical systems, Physica D18 (1986), 280–302.

    MathSciNet  Google Scholar 

  27. Calogero, F.: A class of integrable dynamical systems, Inverse Problems 1 (1985), L21–L24.

    Article  MathSciNet  MATH  Google Scholar 

  28. Calogero, F.: A method to generate solvable nonlinear evolution equations, Lett. Nuovo Cimento 14 (1975), 443–448.

    Article  MathSciNet  Google Scholar 

  29. Calogero, F.: Integrable many-body problems in more than one dimension, Rep. Math. Phys. 24 (1986), 141–143.

    Article  MathSciNet  MATH  Google Scholar 

  30. Calogero, F. and Eckhaus, W.: Nonlinear evolution equations, rescalings, model PDEs and their integrability. I and II, Inverse Problems 3 (1987), 229–262.

    Article  MathSciNet  MATH  Google Scholar 

  31. Calogero, F. and Eckhaus, W.: Nonlinear evolution equations, rescalings, model PDEs and their integrability. I and II, Inverse Problems 4 (1988), 11–33.

    Article  MathSciNet  MATH  Google Scholar 

  32. Papers by Taniuti and others in: Suppl. Prog. Theor. Phys. 55, 1974 (issue devoted to the ‘Reductive perturbation method for nonlinear wave propagation’).

    Google Scholar 

  33. Eckhaus, W.: The long-time behaviour for perturbed wave-equations and related problems, preprint 404, Department of Mathematics, University of Utrecht, December 1985, published in part in Lectures Notes in Physics, Springer, 1986.

    Google Scholar 

  34. Calogero, F.: Universal integrable nonlinear PDEs’, in: P. A. Clarkson (ed.) Applications of Analytical and Geometrical Methods to Nonlinear Differential Equations, NATO ASI Series C: Mathematical and Physical Sciences, Vol. 413, Kluwer, 1993, pp. 109–114.

    Chapter  Google Scholar 

  35. Calogero, F.: C-integrable nonlinear partial differential equations in N+1 dimensions, J. Math. Phys. 33 (1992), 1257–1271.

    Article  MathSciNet  MATH  Google Scholar 

  36. Calogero, F.: Universal C-integrable nonlinear partial differential equation in N+1 dimensions, J. Math. Phys. 34 (1993), 3197–3209.

    Article  MathSciNet  MATH  Google Scholar 

  37. Calogero, F.: The evolution partial differential equation ut = uxxx+3(uxxu2+3u2 xu)+3uxu4, J. Math. Phys. 28 (1987), 538–555.

    Article  MathSciNet  MATH  Google Scholar 

  38. Calogero, F. and De Lillo, S.: The Eckhaus PDE iψt + ψxx + 2(∣ψ∣2)xψ+ ∣ψ∣4ψ = 0, Inverse Problems 3 (1987), 633–681.

    Article  MathSciNet  Google Scholar 

  39. Calogero, F. and De Lillo, S.: The Eckhaus PDE iψt + ψxx + 2(∣ψ∣2)xψ+ ∣ψ∣4ψ = 0, Corrigendum: ibid. 4 (1988) 571.

    MATH  Google Scholar 

  40. Calogero, F.: Interpolation in multidimensions, a convenient finite-dimensional matrix representation of the (partial) differential operators, and some applications, J. Math. Phys. 34 (1993), 2704–4724.

    Google Scholar 

  41. Calogero, F. and Xiaoda, Ji: Solvable (nonrelativistic, classical) n-body problems in multidimensions. I & II, J. Math. Phys. 35 (1994), 710–733

    Article  MathSciNet  MATH  Google Scholar 

  42. M. Costato, A. Degasperis and D. Milani (eds), Nonlinear Dynamics, Conference Proceedings, Vol. 48, Pavullo nel Frignano, Italy, 19–22 May 1994, Soc. Italiana di Fisica, Bologna, 1995.

    Google Scholar 

  43. Solvable (nonrelativistic, classical) n-body problems on the line. I, J. Math. Phys. 34 (1993), 5659–5670.

    Google Scholar 

  44. Arnold, V. I.: Mathematical Methods of Classical Mechanics, 2nd edn, Springer, p. 266.

    Google Scholar 

  45. Jacobi, C.: Problema trium corporum mutis attractionibus cubus distantiarum inverse proportionalibus recta linea se moventium, in: Gesammelte Werke, Bd. IV, Berlin, 1866.

    Google Scholar 

  46. Calogero, F. and Degasperis, A.: Nonlinear evolution equations solvable by the inverse spectral transform. I and II, Nuovo Cimento 32B (1976), 201–242

    MathSciNet  Google Scholar 

  47. Calogero, F. and Degasperis, A.: Nonlinear evolution equations solvable by the inverse spectral transform. I and II, Nuovo Cimento 39B (1977), 1–54.

    MathSciNet  Google Scholar 

  48. Calogero, F. and Degasperis, A.:Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equation, Vol. 1, North-Holland, 1982.

    Google Scholar 

  49. Calogero, F.: A class of C-integrable PDEs in multidimensions, Inverse Problems 10 (1994), 1231–1234.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Fisica, Università di Roma I La Sapienza, 00185, Rome, Italy

    Francesco Calogero

  2. Sezione di Roma, Istituto Nazionale di Fisica Nucleare, Italy

    Francesco Calogero

Authors
  1. Francesco Calogero
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Michiel Hazewinkel Hans W. Capel Eduard M. de Jager

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Calogero, F. (1995). Integrable Nonlinear Evolution Equations and Dynamical Systems in Multidimensions. In: Hazewinkel, M., Capel, H.W., de Jager, E.M. (eds) KdV ’95. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0017-5_12

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-0017-5_12

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4011-2

  • Online ISBN: 978-94-011-0017-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation