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Spectral transform and nonlinear evolution equations

  • F. Calogero
  • A. Degasperis
Part I: Fifteen Review Lectures on Applied Inverse Problems
Part of the Lecture Notes in Physics book series (LNP, volume 85)

Abstract

The spectral transform method for solving nonlinear evolution equations is tersely surveyed.

Keywords

Reflection Coefficient Discrete Spectrum Spectral Problem Nonlinear Evolution Equation Nonlinear Partial Differential Equation 
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.

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References

  1. [1]
    C.S. Gardners J.M. Greene, M.D. Kruskal and R.M. Miura: “Method for solving the Korteweg-de Vries equation”, Phys. Rev. Lett. 19, 1095–1097 (1967).Google Scholar
  2. [2]
    D.J. Korteweg and G. de Vries: “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, 422–443 (1895).Google Scholar
  3. [3]
    V.E. Zakharov and A.B. Shabat: “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media”, Sov. Phys. JETP 34, 62–69 (1972) [Russian original: Zh Eksp. Teor. Fiz. 61, 118-134 (1971)].Google Scholar
  4. [4]
    A.C. Scott, F.Y.F. Chu and D.W. McLanghlin: “The soliton: a new concept in applied science”, Proc. IEEE 61, 1443–1483 (1973).Google Scholar
  5. [5]
    Nonlinear Wave Motion (A.C. Newiell, et.), Lect. Appl. Math. 15, American Mathematical Society, Providence, K.I. 1974.Google Scholar
  6. [6]
    Dynamic al System Theory and Applications (J. Moser, ed.), Lect. Notes in Physics 38, Springer, 1975.Google Scholar
  7. [7]
    Bäcklund Transformations. (R.M. Miura, ed.), Lect. Notes in Mathematics 515, Springer, 1976.Google Scholar
  8. [8]
    Nonlinear Evolution Equations Solvable by the Spectral Transform. (F. Calogerp, ed.), Research Notes in Mathematics, Pitman, 1978.Google Scholar
  9. [9]
    Solitons. (R.K. Bullough, ed.), Lect. Notes in Physics, Springer, 1972.Google Scholar
  10. [10]
    Yu.I. Manin: “Algebraic aspects of nonlinear differential equations”, in Contemporary Problems of Mathematics 11, VINITI, Moscow 1972(in Russian).Google Scholar
  11. [11]
    F. Calogero: “Generalized wronskian relations: a novel approach to Bargmann equivalent and phase-equivalent potentials”, in Studies in Ma- thematical Physics (Essays in honor of Valentine Bargmann), edited by E.H. Lieb, B. Simon and A.S. Wightman, Princeton. University Press, 1976.Google Scholar
  12. [12]
    F. Calogero: “A method to generate solvable nonlinear evolution equations”, Lett. Nuovo Cimento 14, 443–448 (1975).Google Scholar
  13. [13]
    F. Calogero: “Generalized wronskian relations, one-dimensional Schroedinger equation and nonlinear partial differential equations solvable by the inverse-scattering method”, Nuovo Cimento 31B, 229–249 (1976).Google Scholar
  14. [14]
    F. Calogero: “Bäcklund transformations and functional relation for solutions of nonlinear partial differential equations solvable via the inverse-scattering method”, Lett. Nuovo Cimento 14, 537–543 (1975).Google Scholar
  15. [15]
    F. Calogero and A. Degasperis: “Nonlinear evolution. equations solvable by the inverse spectral transform associated with the multichannel Schroedinger problem, and properties of their solutions”, Lett. Nuovo Cimento 15, 65–69 (1976).Google Scholar
  16. [16]
    F. Calogero and A. Degasperis: “Nonlinear evolution equations solvable by the inverse spectral transform.I”, Nuovo Cimento 32B, 201–242 (1976).Google Scholar
  17. [17]
    F. Calogero and A. Degasperis: “Transformations between solutions of different nonlinear evolution equations solvable via the same inverse spectral transform, generalized resolvent formulas and nonlinear operator identites”, Lett. Nuovo Cimento 16, 181–186 (1976).Google Scholar
  18. [18]
    F. Calogero and A. Degasperis: “Coupled nonlinear evolution equations solvable via the inverse spectral transform and solitons that come back: the boomeron”, Lett. Nuovo Cimento 16, 425–433 (1976).Google Scholar
  19. [19]
    F. Calogero and A. Degasperis: “Bäcklund transformations, nonlinear superposition principle, multisoliton solutions and conserved quantities for the “boomeron” nonlinear evolution equation.”, Lett. Nuovo Cimento 16, 434–438 (1976).Google Scholar
  20. [20]
    F. Calogero and A. Degasperis: “Nonlinear evolution equations solvable by the inverse spectral transform.II”, Nuovo Cimento 39B, 1–54 (1977).Google Scholar
  21. [21]
    F. Calogero and A. Degasperis: “Nonlinear evolution. equations solvable by the inverse spectral transform associated to the matrix Schroedinger equation”, in [9].Google Scholar
  22. [22]
    F. Calogero and A. Degasperis: “Special solution of coupled nonlinear evolution equations with bumps that behave as interacting particles”, Lett. Nuovo Cimento 19, 525–533 (1977).Google Scholar
  23. [23]
    F. Calogero: “Nonlinear evolution equations solvable by the inverse spectral transform”, to appear in the Proceedings of the Rome Conference on Mathematical Physics, June 6–15, 1977, edited by G.F. Dell'Antonio, S. Doplicher and G. Jona-Lasimio, Lect. Notes in Physics, Springer, 1978.Google Scholar
  24. [24]
    A. Degasperis: “Solitons, boomerons and trappons”, in [8].Google Scholar
  25. [25]
    A. Degasperis: “Spectral transform and solvability of nonlinear evolution equations; in Proceedings of the Advanced Study Institute on Nonlinear equations in physics and mathematics, August 1977, edited by A.O. Barut, Reidel, 1978.Google Scholar
  26. [26]
    M. Bruschi, D. Levi and O. Raguisco: “Evolution equations associated to the triangular matrix Schroedinger problem solvable by the inverse spectral transform”, Nuovo Cimento (in press).Google Scholar
  27. [27]
    K.M. Case and S.C. Chu: “Some remarks on the wronskian technique and the inverse scattering transform”, J. Math. Phys. 18, 2044–2052 (1977).Google Scholar
  28. [28]
    S.C. Cru and J.F. Ladik: “Generating exactly soluble nonlinear discrete evolution equations by a generalized wronskian technique”, J. Math. Phys. 18, 690–700 (1977).Google Scholar
  29. [29]
    F. Calogero and A. Degasperis: “Extension of the spectral transform method for solving nonlinear evolution equations”, Lett. Nuovo Cimento.Google Scholar
  30. [30]
    F. Calogero and A. Degasperis: “Exact solution via the spectral transform of a nonlinear evolution equation with linear x-dependent coeffi cients”, Lett. Nuovo CimentoGoogle Scholar
  31. [31]
    F. Calogero and A. Degasperis: “Extension of the spectral transform method for solving nonlinear evolution equations.II”, Lett. Nuovo CimentoGoogle Scholar
  32. [32]
    F. Calogero and A. Degasperis: “Exact solution via the spectral transform of a generalization with linearly x-dependent coefficients of the modified Korteweg-de Vries equations”, Lett. Nuovo CimentoGoogle Scholar
  33. [33]
    F. Calogero and A. Degasperis: “Exact solution via the spectral transform of a generalisation with linearly x-dependent coefficients of the nonlinear Schroedinger equation”, Lett. Nuovo CimentoGoogle Scholar
  34. [34]
    F. Calogero and A. Degasperis: “Conservation laws for classes of nonlinear evolution equations solvable by the spectral transform”, Commun. Math. Phys. (submitted to).Google Scholar
  35. [35]
    D. Levi and O. Reguisco: “Extension of the spectral transform method for sol ving nonlinear differential difference equations”, Lett. Nuovo Cimento (in press).Google Scholar
  36. [36]
    P.D. Lax: “Integrals of nonlinear equations of evolution, and solitary waves”, Comm. Pure Appl. Math. 21, 467–490 (1968).Google Scholar
  37. [37]
    J.C. Eilbeck: “Boomeron”, a computer-produced film (Mathematics Dept., Heriot Watt University, Edinburgh).Google Scholar
  38. [38]
    J.C. Eilbeck: “Zoomerons”, a computer-produced film (Mathematics Dept., Heriot-Watt University, Edinburgh).Google Scholar
  39. [39]
    F. Calogero and A. Degasperis (to be published).Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • F. Calogero
    • 1
    • 2
  • A. Degasperis
    • 1
    • 2
  1. 1.Istituto di Fisica dell'Universita di RomaRoma
  2. 2.Istituto Nazionale di Fisica Nucleare Sezione di RomaRoma

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