Some Ideas on Nonlinear Evolution Equations

  • F. Calogero
Conference paper
Part of the Research Reports in Physics book series (RESREPORTS)


This is a terse review of some ideas and recent results helpful to understand nonlinear evolution equations. The discussion is mainly limited to problems in 1+1 dimensions.




Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. (1).
    F. Calogero and W. Eckhaus: “Nonlinear evolution equations, rescalings, model PDEs and their integrability. I & II”. Inverse Problems 3, 229–262 (1987) & 4, 11–33 (1988).Google Scholar
  2. (2).
    F. Calogero: “The evolution PDE ut, = uxxx + 3 (uxx u2 + 3 ux2 u) + 3 ux u4 ”. J. Math. Phys. 28, 538–555 (1987).Google Scholar
  3. (3).
    F. Calogero and S. De Lillo: “The Eckhaus PDE i psit + psixx + 2 (\psi\2)x psi \psi\4 psi = 0 ”. Inverse Problems 3, 633–681 (1987).Google Scholar
  4. (4).
    F. Calogero and S. De Lillo: “Cauchy problem on the semiline and on a finite interval for the Eckhaus equation”. Inverse Problems 4, L33 - L37 (1988).Google Scholar
  5. (5).
    F. Calogero and S. De Lillo: “The Burgers equation on the semi-infinite and finite intervals”. Nonlinearity 2, 37–43 (1989).Google Scholar
  6. (6).
    F. Calogero and S. De Lillo: (to be published).Google Scholar
  7. (7).
    Galileo Galilei: j1 Saggiatgre, 1623. CSee: Opere di Galileo Galilei, Edizione nazionale, Barbera, Firenze, 1890–1909, vol. VI, p. 2327.Google Scholar
  8. (8).
    F. Calogero: “Why are certain nonlinear PDEs both widely applicable and integrable?”. Rome preprint n. 582, January 1988. To be published in: V. E. Zakharov (editor): Wheat is integrability (for nonlinear PDEs)?, Springer, 1989 (in press).Google Scholar
  9. (9).
    F. Calogero and W. Eckhaus: “Necessary conditions for integrability of nonlinear PDEs”. Inverse Problems 3, L27 - L32 (1987).Google Scholar
  10. (10).
    F. Calogero and A. Maccari: “Equations of nonlinear Schroedinger type in 1+1 and 2+1 dimensions, obtained from integrable PDEs”. In: P. C. Sabatier (editor): Inverse Problems: an Interdisciplinary Study (Proceedings of the meeting on Inverse Problems held in Montpellier, November 1986), Advances in Electronics and Electron Physics, 19, Academic Press, London & New York, 1987, pp. 463–480.Google Scholar
  11. (11).
    F. Calogero: “Universality and integrability of the nonlinear PDEs describing N-wave interactions”. J. Math. Phys. 30, 28–40 (1989).Google Scholar
  12. (12).
    F. Calogero: “Solutions of certain integrable nonlinear evolution PDEs describing nonresonant N-wave interactions”. J. Math Phys. (in press).Google Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1990

Authors and Affiliations

  • F. Calogero
    • 1
    • 2
  1. 1.Dipartimento di FisicaUniversità di Roma “La Sapienza”RomaItaly
  2. 2.Sezione di RomaIstituto Nazionale di Fisica NucleareItaly

Personalised recommendations