Nonlinear wave equations

  • Martin Kruskal
III. Nonlinear Differential Equations
Part of the Lecture Notes in Physics book series (LNP, volume 38)


Solitary Wave Nonlinear Wave Equation High Order Equation Linear Wave Equation Galilean Invariance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


The main work of our group is distilled in a sequence of papers referred to in the text simply by the corresponding roman numeral. Those that have appeared by now are

  1. Miura, R. M., Korteweg-deVries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys. 9, 1 (1968), 1202–1204.Google Scholar
  2. Miura, R. M., Gardner, C. S., and Kruskal, M. D., Korteweg-deVries equation and generalizations. II. Existence of conservation laws and constants of motion. J. Math. Phys. 9 (1968), 1204–1209.Google Scholar
  3. Su, C. H., and Gardner, C. S., Korteweg-deVries equation and generalizations. III. Derivation of the Korteweg-deVries equation and Burgers' equation. J. Math. Phys 10 (1969), 536–539.Google Scholar
  4. Kruskal, M. D., Miura, R. M., Gardner, C. S., and Zabusky, N.J., Korteweg-deVries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws, J. Math. Phys., 11 (1970), 952–960.Google Scholar
  5. Gardner, C. S., Greene, J. M., Kruskal, M.D., and Miura, R.M., Korteweg-deVries equation and generalizations. VI. Methods for exact solution, Comm. Pure Appl. Math. 27 (1974), 97–133.Google Scholar

The initial discovery of solitons, and of the method of exact solution of the KdV equation, were announced respectively in

  1. Zabusky, N. J., and Kruskal, M. D., Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys. Rev. Letters 15 (1965), 240–243.Google Scholar
  2. Gardner, C. S., Greene, J. M., Kurskal, M. D., and Miura, R.M., Method for solving the Korteweg-deVries equation, Phys. Rev. Letters 19 (1967), 1095–1097.Google Scholar

The remaining works referred to in the text are

  1. Benjamin, T. B., Bona, J. L., and Mahony, J. J., Model equations for long waves in nonlinear dispersive systems, Phil. Trans. Roy. Soc. London A, 272 (1972), 47–78.Google Scholar
  2. Berger, N., Estimates for the derivatives of the velocity and pressure in shallow water flow and approximate shallow water equations, SIAM J. Appl. Math. 27 (1974), 256–280; see especially p. 276.Google Scholar
  3. Cole, J. D., On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math., 9 (1951), 225–236.Google Scholar
  4. Fermi, E., Pasta, J. R., and Ulam, S. M., Studies of nonlinear problems, part I, Report LA-1940, Los Alamos Scientific Laboratory, Los Alamos, N. M., (1955).Google Scholar
  5. Hopf, E., The partial differential equation ut + uux = μuxx ' Comm. Pure Appl. Math. 3 (1950), 201–230.Google Scholar
  6. Korteweg, D. J., and deVries, 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.Google Scholar
  7. Kruskal, M. D., The Korteweg-deVries equation and related evolution equations, Amer. Math. Soc. Lectures in Appl. Math. 15 (1974), 61–83.Google Scholar
  8. Lax, P. D., Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467–490.Google Scholar
  9. Rubinstein, J., Sine-Gordon equation, J. Math. Phys. 11 (1970) 258–266.Google Scholar
  10. Témam, R., Sur un probleme nonlinéaire J. Math. Pures Appl., 48 (1969), 159.Google Scholar
  11. Thickstun, W. R., A system of particles equivalent to solitons, to be published.Google Scholar
  12. Whitham, G. B., Non-linear dispersive waves, Proc. Roy. Soc. A, 283 (1965), 238–261.Google Scholar
  13. Zakharov, V. E., and Shabat, A. B., Exact theory of twodimensional self-focussing and one-dimensional self-modulating of waves in nonlinear media, English translation in Soviet Physics JETP 34 (1972), 62–69.Google Scholar

Copyright information

© Springer-Verlag 1975

Authors and Affiliations

  • Martin Kruskal
    • 1
  1. 1.Department of Astrophysical SciencesPrinceton UniversityN.J.

Personalised recommendations