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The inverse scattering transform for multidimensional (2+1) problems

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Nonlinear Phenomena

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References

  1. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitayevskiî, Theory of Solitons. The Method of the Inverse Scattering Problem, Nauka, Moscow (in Russian), 1980.

    Google Scholar 

  2. R. Beals and R. R. Coifman:(i) Scattering, transformations spectrales, et équations d'évolution non lineaires, Seminaire Goulaoric-Meyer-Schwartz, 1980–1981, exp. 22, École Polytechnique, Palaiseau; (ii) Scattering and inverse scattering for first order systems (preprint);(iii) Scattering, transformations spectrales, et équations d'évolution non lineaires, Seminaire Goulaoric-Meyer-Schwartz, 1981–1982, exp. 21, École Polytechnique, Palaiseau.

    Google Scholar 

  3. (i) I. Hauser and F. J. Ernst, A homogeneous Hilbert problem for the Kinnersley-Chitre transformations. J. Math. Phys. 21, 1126–1140 (1980);(ii) A homogeneous Hilbert problem for the Kinnersley-Chitre transformations of electrovac space-times. ibid. 21, 1418–1422 (1980); (iii) Proof of a Geroch conjecture. ibid. 22, 1051 (1981); (iv) V. A. Belinskiî and V. E. Zakharov, Integration of the Einstein equations by the inverse scattering method and calculation of the exact soliton solution. Sov. Phys. JETP 48, 985993 (1978); (v) C. Cosgrove, Bäcklund transformations in the Hauser-Ernst formalism for stationary axisymmetric space-times. J. Math. Phys. 22, 2624–2639 (1981).

    Article  Google Scholar 

  4. [4](i) M. F. Atiyah and R. S. Ward, instantons and algebraic geometry. Commun. Math. Phys. 66, 117–124 (1977); (ii) M. F. Atiyah, N. J. Hitchin, V. G. Drinfield, and Manin I. Yu., Construction of instantons. Phys. Lett. 65A, 185–187 (1978).

    Article  Google Scholar 

  5. P. J. Olver, Evolution equations possessing infinetely many symmetries J. Math. Phys. 18, 1212–1215 (1977); Math. Proc. Camb. Phil. Soc. 88, 71 (1980).

    Article  Google Scholar 

  6. F. Magri, A simple model of the integrable hamiltonian equation. J. Math. Phys. 19, 1156–1162 (1978); ibid. Lecture Notes in Physics #120, p. 233, M. Boiti, F. Pipinelli, and G. Soliani, eds. Springer Verlag (1978).

    Article  Google Scholar 

  7. (i) I. M. Gel'fand and L. A. Dikii, Resolvents and hamiltonian systems. Funct. Anal. Appl. 11, 93–104 (1977);(ii) I. M. Gel'fand and I. Ya. Dorfman, Funct. Anal. Appl. 13, 248 (1979); ibid. 14 (1980).

    Article  Google Scholar 

  8. A. S. Fokas, A symmetry approach to exactly solvable evolution equations. J. Math. Phys. 21, 1318 (1980)

    Article  Google Scholar 

  9. A. S. Fokas and R. L. Anderson, On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for hamiltonian systems. J. Math. Phys. 23, 1066 (1982)

    Article  Google Scholar 

  10. A. S. Fokas and B. Fuchssteiner, Bäcklund transformations for hereditary symmetries. Nonlinear Anal. TMA 5, 423 (1981)

    Article  Google Scholar 

  11. —ibid., On the structure of symplectic operators and hereditary symmetries. Lett. Nuovo Cimento 28, 299 (1980)

    Google Scholar 

  12. —ibid., The hierarchy of the Benjamin-Ono equation. Phys. Lett. 86A, 341 (1981)

    Google Scholar 

  13. B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physics 4D, 47 (1981)

    Google Scholar 

  14. B. Fuchssteiner, Nonlinear Anal. TMA 3, 849 (1979); ibid. Prog. Theor. Phys. 65, 861 (1981); ibid. 68 (1982); ibid. Lett. Math. Phys. 4, 1977 (1980).

    Google Scholar 

  15. A. P. Fordy and J. Gibbons, Factorization of operators. II. J. Math. Phys. 22, 1170 (1981); ibid., Integrable nonlinear Klein-Gordon equations and Toda lattices. Commun. Math. Phys. 77, 21 (1980).

    Google Scholar 

  16. (i) B. A. Kupersbmidt and G. Wilson, Invent. Math. 62, 403 (1981); ibid. Commun. Math. Phys. 81, 189 (1981); (ii) Y. Kosmann-Schwartzbach, Hamiltonian systems of fibered manifolds. Lett. Math. Phys. 5, 229–237 (1981).

    Google Scholar 

  17. B. G. Konopelchen'ko, Transformation properties of the integrable evolution equations. Phys. Lett. 75A, 447 (1980); ibid. 79A, 39 (1980); ibid. 95B, 83 (1980); ibid. 100B, 254-260 (1981).

    Google Scholar 

  18. G. Z. Tu, Commutativity theorem of partial differential equations. Commun. Math. Phys. 77, 289–297 (1980).

    Google Scholar 

  19. (i) C. S. Gardner, The Korteweg-de Vries equation and generalizations. IV. The Korteweg-de Vries equation as a hamiltonian system. J. Math. Phys. 12, 1548–1551 (1971); (ii) V. E. Zakharov and L. D. Fadeev, Korteweg-de Vries equation, a completely integrable hamiltonian system. Funct. Anal. Appl. 5, 280–287 (1971).

    Google Scholar 

  20. H. D. Wahlquist and F. B. Estabrook, Prolongation structures and nonlinear evolution equations J. Math. Phys. 16, 1–7 (1975 ibid. 17, 1293–1297 (1976); see also the contribution of these authors in Nonlinear Equations Solvable Via the IST, F. Calogero ed. (1977).

    Article  Google Scholar 

  21. L. Abellanas and A. J. Galindo, J. Math. Phys. 20, 1239 (1979);ibid. J. Math. Phys.24, 504 (1983); ibid. Lett. Math. Phys. 6, 391 (1982).

    Google Scholar 

  22. W. Oevel and B. Fuchssteiner, Explicit formulas for symmetries and conservation laws of the Kadomtsev-Petviashvili equation. Phys. Lett. 88A, 323–327 (1982).

    Google Scholar 

  23. H. H. Chen, J. C. Lee, and J. E. Lin, to appear in Nonlinear Waves (preprint 1982).

    Google Scholar 

  24. B. G. Konopelchen'ko, The two-dimensional matrix spectral problem: General structure of the integrable equations and their Backlund transformations. Phys. Lett. 86A, 346–350 (1981); ibid. Commun. Math. Phys. (1982); ibid. J. Phys. A. 15 (1982).

    Google Scholar 

  25. M. Kashiwara and T. Miwa, Proc. Japan Acad. 57 A, 342 (1981); E. Date, M. Kashiwara, and T. Miwa, Proc. Japan Acad. 57 A, 387 (1981); M. Jimbo, E. Date, M. Kashiwara, and T. Miwa, J. Phys. Soc. of Japan 50, 3806 (1981); and RIMS preprints 359 and 360.

    Google Scholar 

  26. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM Studies in Applied Mathematics #4, 1981.

    Google Scholar 

  27. H. H. Chen, A Bäcklund transformation in two dimensions. J. Math. Phys. 16, 2382–2384 (1975); D. V. Chudnovski, J. Math. Phys. 20, 2416 (1979).

    Google Scholar 

  28. J. Satsuma, N-soliton solution of the two-dimensional Korteweg-de Vries equation. J. Phys. Soc. of Japan 40, 286–290 (1976);J. Satsuma and M. J. Ablowitz, Two-dimensional lumps in nonlinear dispersive systems. J. Math. Phys. 20, 1496 (1979); A. Nakamura, Phys. Rev. Lett. 46, 751 (1981); ibid. Phys. Lett. 88A, 55 (1982); ibid. J. Math. Phys. 22, 2456 (1981); ibid. J. Phys. Soc. of Japan 50, 2469 (1981) ibid. 51, 19 (1981).

    Google Scholar 

  29. V. E. Zakharov and P. B. Shabat, A scheme for integrating the nonlinear equations cf mathematical physics by the method of the inverse scattering problem. I. Funct. Anal. Appl. 8, 226–235 (1974).

    Article  Google Scholar 

  30. (i) V. E. Zakharov and P. B.Shabat, Integration of the nonlinear equations of mathematical physics by the method of the inverse scattering problem. II. Funct. Anal. Appl. 13, 166–174 (1979);(ii) V. E. Zakharov and A. V. Mikhaîlov, Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method. Sov. Phys. JETP 47, 1071–1027 (1978).

    Google Scholar 

  31. S. V. Manakov (private communication).

    Google Scholar 

  32. A. S. Fokas and M. J. Ablowitz, On a linearization of the Korteweg-de Vries and Painlevé II equations. Phys. Rev. Lett. 47, 1096 (1981).

    Article  Google Scholar 

  33. P. Santini, M. J. Ablowitz, and A. S. Fokas, The Direct Linearization and the RH Direct Method. (Preprint).

    Google Scholar 

  34. R. Rosales, Exact solutions of some nonlinear evolution equations. Stud. Appl. Math. 59, 117–151 (1978).

    Google Scholar 

  35. M. J. Ablowitz, A. S. Fokas, and R. L. Anderson, The direct linearizing transform and the Benjamin-Ono equation. Phys. Lett. A. 93, 375 (1983).

    Article  Google Scholar 

  36. A. S. Fokas and M. J. Ablowitz, The inverse scattering transform for the Benjamin-Ono —a pivot to multidimensional problems. Stud. Appl. Math. 68, 1–10 (1983).

    Google Scholar 

  37. R. G. Newton, J. Math. Phys. 21, 493 (1980); ibid. Geophys. J. R. Astr. Soc. 65, 191 (1981).

    Article  Google Scholar 

  38. (i) G. R. W. Quispel and H. W. Capel, Phys. Lett. 88A, 371 (1981); ibid 85A, 248 (1981); ibid Physics 110A, 41 (1981); (ii) F. W. Nijhoff, J. Van der Linden, G. R. W. Quispel, H. W. Capel, and J. Velthuizen, Physics A, (1982).

    Google Scholar 

  39. G. B. Whitham (private communication).

    Google Scholar 

  40. A. S. Fokas and M. J. Ablowitz, On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23, 2033–2042 (1982).

    Article  Google Scholar 

  41. A. S. Fokas and M. J. Ablowitz, Direct Linearization of the Korteweg-de Vries Equation. AIP Conference Proceedings #88, 237–241 (1982). M. Tabor and Y. M. Treve eds.

    Google Scholar 

  42. A. S. Fokas and M. J. Ablowitz, On the inverse scattering and direct linearizing transforms for the Kadomtsev-Petviashvili equation. Phys. Lett. A. 94A, 67–70 (1983).

    Article  Google Scholar 

  43. M. J. Ablowitz and A. S. Fokas, A Direct Linearization Associated with the Benjamin-Ono equation. Conference Proceedings #88, 229–236 (1982). M. Tabor and Y. M. Treve eds.

    Google Scholar 

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

    Google Scholar 

  45. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, The Korteweg-de Vries equation and generalizations. VI. Methods for exact solution. Comm. Pure Appl. Math. 27, 97–133 (1974).

    Google Scholar 

  46. P. Delft and E. Trubowitz, Inverse scattering on the line. Comm. Pure Appl. Math. 32, 121–125 (1979).

    Google Scholar 

  47. (i) M. J. Ablowitz, D. J. Kaup, A. C. Newel, and H. Segur, Method for solving the sine-Gordon equation. Phys. Rev. Lett. 30, 1262–1264 (1973); (ii) ibid., Nonlinear evolution equations of physical significance. Phys. Rev. Lett. 31, 125–127 (1973);(iii) ibid., The inverse scattering transform —Fourier analysis for nonlinear problems. Stud. Appl. Math. 53, 249–315 (1974).

    Article  Google Scholar 

  48. V. E. Zakharov and P. 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).

    Google Scholar 

  49. G. B. Whitham, Linear and Nonlinear Waves. Wiley-Interscience, 1974.

    Google Scholar 

  50. D. J. Kaup, On the inverse scattering problem for cubic eigenvalue problems of the class ψ xxx + 6Qψ x + 6Rψ = λψ. Stud. Appl. Math. 62, 189 (1980).

    Google Scholar 

  51. P. Delft, C. Tomei, and E. Trubowitz, Commun. Pure Appl. Math. 35, 567 (1982).

    Google Scholar 

  52. D. J. Kaup, The three-wave interaction —a nondispersive phenomenon. Stud. Appl. Math. 55, 9–44 (1976).

    Google Scholar 

  53. V. E. Zakharov and S. V. Manakov, The theory of resonance interaction of wave packets in nonlinear media. Sov. Phys. JETP 42, 842–850 (1976).

    Google Scholar 

  54. W. Symes, Relations among generalized Korteweg-de Vries systems. J. Math. Phys. 20, 721–725 (1979).

    Article  Google Scholar 

  55. M. J. Ablowitz and R. Haberman, Nonlinear evolution equations —two and three dimensions. Phys. Rev. Lett. 35, 1185–1188 (1975).

    Article  Google Scholar 

  56. A. C. Newell, The general structure of integrable evolution equations. Proc. R. Soc. London A365, 283–311 (1979).

    Google Scholar 

  57. R. Beals, The Inverse Problem for Ordinary Differential Operators on the Line, Yale, 1982 (preprint).

    Google Scholar 

  58. P. J. Caudrey, The inverse problem for a general N X N spectral equation. Physics 6D, 51–66 (1982).

    Google Scholar 

  59. (i) F. Calogero and A. Degasperis, Nonlinear evolution equations solvable by the inverse spectral transform. I and II. Nuovo Cimento 32B, 201–242 (1976); (ii) ibid., Nonlinear evolution equations solvable by the inverse spectral transform. Nuovo Cimento 39B, 1–54 (1977).

    Google Scholar 

  60. (i) M. Wadati, The modified Korteweg-de Vries equation. J. Phys. Soc. of Japan 32, 1681 (1977); (ii) M. Wadati, H. Sanki, and K. Konno, Relationships among inverse methods, Bäcklund transformation and an infinite number of conservation laws. Prog. Theor. Phys. 53, 419–436 (1975).

    Google Scholar 

  61. M. Jaulent, Inverse scattering problems in absorbing media. J. Math. Phys. 17, 1351–1360 (1976).

    Article  Google Scholar 

  62. D. J. Kaup and A. C. Newell, An exact solution for a derivative nonlinear Schrödinger equation. J. Math. Phys. 19, 798–801 (1978); A. V. Mikhaîlov, Sov. Phys. JETP 23, 356 (1976).

    Article  Google Scholar 

  63. L. M. Alonso, Schrödinger spectral problems with energy-dependent potentials as sources of nonlinear Hamiltonian evolution equations. J. Math. Phys. 21, 2342–2349 (1980).

    Article  Google Scholar 

  64. (i) Y. Kodama, J. Satsuma, and M. J. Ablowitz, Nonlinear intermediate long wave equation: analysis and method of solution. Phys. Rev. Lett. 46, 687–690 (1981);(ii) Y. Kodama, M. J. Ablowitz, and J. Satsuma, Direct and inverse scattering problem of nonlinear intermediate long wave equations. J. Math. Phys. 23, 564–576 (1982).

    Article  Google Scholar 

  65. T. L. Bock and M. D. Kruskal, A two-parameter Miura transformation of the Benjamin-Ono equation. Phys. Lett. 74A, 173–176 (1979).

    Google Scholar 

  66. A. Nakamura, A direct method calculating periodic wave solutions to nonlinear evolution equations. I. Exact two-periodic wave solution. J. Phys. Soc. of Japan 47, 1701–1706 (1979).

    Google Scholar 

  67. J. Satsuma, T. Taha, and M. J. Ablowitz, On a Bäcklund Transformation and Scattering Problem for the Modified Intermediate Long Wave Equation. Preprint INS #12, May 1982.

    Google Scholar 

  68. E. Hille, ODE'S in the Complex Domain. Wiley-Interscience, 1976.

    Google Scholar 

  69. B. Riemann, Gesammelte Mathematische Werke. Leipzig, 1982.

    Google Scholar 

  70. D. V. Chudnovski, In Bifurcation Phenomena in Mathematical Physics and Related Topics, p. 386. C. Bardos and D. Bessis eds., D. Reidel, 1980.

    Google Scholar 

  71. H. Flaschka and A. C. Newell, Monodromy and spectrum preserving deformations. Comm. Math. Phys. 76, 65–116 (1980).

    Article  Google Scholar 

  72. (i) M. Sato, T. Miwa, and M. Jimbo, RIMS, Kyoto Univ. 14, 223 (1977); ibid. 15, 201 (1979); ibid. 15, 871 (1979);(ii) M. Ambo, T. Miwa, and K. Ueno, Physica 2D, 306 (1981); M. Jimbo and T. Miwa, Physica, 2D, 407 (1981); ibid. Physica 4D, 26 (1982).

    Google Scholar 

  73. D. Hilbert, Grundzüge der Integralgleichungen. Leipzig, 1924.

    Google Scholar 

  74. J. Plemelj, Problems in the Sense of Riemann and Klein. Wiley, 1964.

    Google Scholar 

  75. G. D. Birkhoff, Transactions of the AMS, 436 (1909); ibid. 199 (1910).

    Google Scholar 

  76. N. P. Erugin, Linear Systems of Ordinary Differential Equations. Academic Press, 1966.

    Google Scholar 

  77. F. D. Gakhov, Boundary Value Problems. Pergamon, 1966.

    Google Scholar 

  78. N. I. Muskhelishvili, Singular Integral Equations. Noordhoff, Groningen, 1953.

    Google Scholar 

  79. N. P. Vekua, Systems of Singular Integral Equations. Gordon and Breach, 1967.

    Google Scholar 

  80. (i) P. P. Zabreyko, A. I. Koshelev, M. A. Krasnoselskiî, S. G. Mikhlin, L. S. Rakovshchik, and V. Ya, Integral Equations —a Reference Text. Noordhoff Int., Leyden, 1975; (ii) S. Prossdorf, Some Classes of Singular Equations. North-Holland, 1978.

    Google Scholar 

  81. M. G. Krein, Uspekhi Mat. Nauk 13, 3 (1958).

    Google Scholar 

  82. I. Gohberg and M. G. Krein, Uspekhi Mat. Nauk 13, 2 (1958).

    Google Scholar 

  83. P. Delft (private communication).

    Google Scholar 

  84. A. S. Fokas and M. J. Ablowitz, On the initial value problem of Painlevé II. (Preprint).

    Google Scholar 

  85. R. G. Newton, On a generalized Hilbert problem. J. Math. Phys. 23, 2257–2265 (1982).

    Article  Google Scholar 

  86. B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersive media. Sov. Phys. Doklady 15, 539–541 (1970).

    Google Scholar 

  87. V. Dryuma, Sov. Phys. JETP Lett. 19, 387 (1974).

    Google Scholar 

  88. A. Davey and K. Stewartson, On three-dimensional packets of surface waves. Proc. Roy. Soc. London A338, 101–110 (1974).

    Google Scholar 

  89. D. J. Benney and G. J. Roskes, Wave instabilities. Stud. Appl. Math. 48, 377–385 (1969).

    Google Scholar 

  90. (i) V. E. Zakharov and S. V. Manakov, Soliton theory. Sov. Sci. Phys. Rev. 1, 133–190 (1979); (ii) S. V. Manakov, The inverse scattering transform for the time-dependent Schrödinger equation and Kadomtsev-Petviashvili equation. Physica 3D, 420–427 (1981).

    Google Scholar 

  91. H. Segur, Comments on IS for the Kadomtsev-Petviashvili Equation, AIP Conference Proceedings #88, 211–228 (1982). La Jolla, 1981. M. Tabor and Y. M. Treve eds.

    Google Scholar 

  92. (i) D. J. Kaup, The inverse scattering solution for the full three-dimensional three-wave resonant interaction. Physica 1D, 45–67 (1980); (ii) H. Cornille, Solutions for the nonlinear three-wave equations in three-spatial dimensions. J. Math. Phys. 20, 1653–1666 (1979); (iii) L. P. Niznik, Ukranian Math. J. 24, 110 (1972).

    Google Scholar 

  93. A. S. Fokas and M. J. Ablowitz, On the inverse scattering of the time dependent Schrödinger equation and the associated KPI equation, INS #22, Nov. 1982 (to appear in Stud. Appl. Math.).

    Google Scholar 

  94. M. J. Ablowitz, D. Bar Yaacov, and A. S. Fokas, On the IST for KPII, INS #21, Nov. 1982 (to appear in Stud. Appl. Math.).

    Google Scholar 

  95. A. S. Fokas, On the inverse scattering of first order systems in the plane related to nonlinear multidimensional equations, INS #23, Dec. 1982 (preprint).

    Google Scholar 

  96. A. S. Fokas and M. J. Ablowitz, On a method of solution for a class of multidimensional problems. (Preprint).

    Google Scholar 

  97. A. S. Fokas and M. J. Ablowitz, On the inverse scattering transform of multidimensional nonlinear equations related to first order systems in the plane. Preprint INS #24, Jan. 1983.

    Google Scholar 

  98. L. Hörmander, Complex Analysis in Several Variables. Van Nostrand, 1966.

    Google Scholar 

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Fokas, A.S., Ablowitz, M.J. (1983). The inverse scattering transform for multidimensional (2+1) problems. In: Wolf, K.B. (eds) Nonlinear Phenomena. Lecture Notes in Physics, vol 189. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-12730-5_6

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