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Reduction technique for matrix nonlinear evolution equations

  • A. Degasperis
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 120)

Keywords

Reflection Coefficient Nonlinear Evolution Equation Nonlinear Partial Differential Equation MKdV Equation Inverse Scattering Method 
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]
    GARDNER, C.S., GREENE, J.M., KRUSKAL, M.D., MIURA, R.M.: “Method for solving the Korteweg-de Vries equation”. Phys. Rev. Lett. 19, 1095–1097 (1967).CrossRefGoogle Scholar
  2. [2]
    a) ZAKHAROV, V.E., SHABAT, A.B.: “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media”. Sov. Phys. JETP 34, 62–69 (1972) [Zh.Eksp.Teor.Fiz. 161, 118(1971); b) ABLOWITZ, M.J., KAUP, D.J., NEWELL, A.C., SEGUR, H.: “The inverse scattering transform-Fourier analysis for nonlinear problems”. Studies Appl.Math. 53, 249–315(1974); c) CALOGERO, F., DEGASPERIS, A.: “Nonlinear evolution equations solvable by the inverse spectral transform, I”. Nuovo Cimento 32B, 201–242 (1976); d) CALOGERO, F., DEGASPERIS, A.: “Nonlinear evolution equations solvable by the inverse spectral transform, II”. Nuovo Cimento 39B, 1–54(1977); NEWELL; A.C.: “The general structure of integrable evolution equations”. Proc. R. Soc. Lond. A 365, 283–311(1979) CALOGERO, F., DEGASPERIS, A.: “Extension of the Spectral Transform method for solving nonlinear evolution equations, I & II”. Lett.Nuovo Cimento 22, 131–137(1978) & 22, 263–269(1978); CALOGERO, F., DEGASPERIS, A.: “Solution by the Spectral Transform method of a nonlinear evolution equation including as a special case the Cylindrical KdV Equation”. Lett. Nuovo Cimento 23, 150–154(1978).Google Scholar
  3. [3]
    BULLOUGH, R.K., CAUDREY, P.J.: “The soliton and its history” in “Solitons” (Bullough R.K. and Caudrey, P.J., eds), Lecture Notes in Physics, Springer, Heidelberg, 1979; and also references quoted there.Google Scholar
  4. [4]
    WADATI, M., KAMIJO, T.: “On the extension of inverse scattering method”. Prog. Theor.Phys. 52, 397–414 (1974). Ref. (2-d). ZAKHAROV, V.E.:“The inverse scattering method”, in “Solitons” (Bullough, R.K. and Caudrey, P.J., eds.), Lecture Notes in Physics, Springer, Heidelberg, 1979.Google Scholar
  5. [5]
    CALOGERO, F., DEGASPERIS, A.: “Reduction technique for matrix nonlinear evolution equations solvable by the Spectral Transform”. To appear in J. of Math.Phys.Google Scholar
  6. [6]
    For an introductory review, see DEGASPERIS, A.: “Spectral Transform and solvability on nonlinear evolution equations” in “Nonlinear Problems in Theoretical Physics” (Rañada, A.F. ed.) Lecture Notes in Physics, 98, Springer 1979, pp.35–90.Google Scholar
  7. [7]
    The class of solvable equations is actually larger, as a t-dependent polynomial of L could act also on Qt and the functions n and β v could dependent also on t. See Ref. (2-d).Google Scholar
  8. [8]
    DEGASPERIS, A.: “Solitons, Boomerons, Trappons”, in “Nonlinear Evolution Equations solvable by the Spectral Transform” (Calogero F., ed.) Research Notes in Mathematics 26, Pitman Publishing, London, 1978, pp.97–126.Google Scholar
  9. [9]
    CALOGERO, F., DEGASPERIS, A.: “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
  10. [10]
    JAULENT, M., LEON, J.J.P.: “Nonlinear evolution equations associated with a massive Dirac system”. Lett.Nuovo Cimento 23, 137 (1978).Google Scholar
  11. [11]
    BAUSCHI, M., LEVI, D., RAGNISCO, O.: “Nonlinear evolution equations solvable by the inverse Spectral Transform associated to the matrix Schrödinger equation of rank 4”. Nuovo Cimento 43B, 251–270 (1978).Google Scholar
  12. [12]
    CALOGERO, F., DEGASPERIS, A.: “Conservation laws for classes of nonlinear evolution equations solvable by the Spectral Transform”. Comm.Math.Phys. 63, 155–176 (1978).Google Scholar
  13. [13]
    HIROTA, R.: “Exact envelope-soliton solutions of a nonlinear wave equation”. J.Math.Phys. 14, 805 (1973).Google Scholar
  14. [14]
    PIRANI, F., SOLIANI, G. (private communication).Google Scholar
  15. [15]
    Private communication.Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • A. Degasperis
    • 1
    • 2
  1. 1.Istituto di FisicaUniversità di RomaRomaItaly
  2. 2.Istituto Nazionale di Fisica NucleareSezione di RomaItaly

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