Skip to main content

The Inverse Scattering Problem for the Zakharov–Shabat System

  • Chapter
Integrable Hamiltonian Hierarchies

Part of the book series: Lecture Notes in Physics ((LNP,volume 748))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur. The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math., 53:249–315, 1974.

    MathSciNet  Google Scholar 

  2. V. I. Karpman and E. M. Maslov. Perturbation theory for solitons. Sov. Phys. JETP, 46(2):281–291, 1977.

    ADS  MathSciNet  Google Scholar 

  3. V. I. Karpman and V. V. Solov’ev. A perturbational approach to the two-soliton systems. Physica D: Nonl. Phen., 3(3):487–502, 1981.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. V. S. Gerdjikov, D. J. Kaup, I. M. Uzunov, and E. G. Evstatiev. Asymptotic behavior of N-soliton trains of the nonlinear Schrödinger equation. Phys. Rev. Lett., 77(19):3943–3946, 1996.

    Article  ADS  Google Scholar 

  5. V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, and G. L. Diankov. Nonlinear Schrödinger equation and N-soliton interactions: Generalized Karpman-Solov’ev approach and the complex Toda chain. Phys. Rev. E, 55(5):6039–6060, 1997.

    Article  ADS  MathSciNet  Google Scholar 

  6. V. S. Gerdjikov, E. G. Evstatiev, D. J. Kaup, G. L. Diankov, and I. M. Uzunov. Stability and quasi-equidistant propagation of NLS soliton trains. Phys. Lett. A, 241(6):323–328, 1998.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. M. Toda. Waves in nonlinear lattice. Suppl. Prog. Theor. Phys., 45:174–200, 1970.

    Article  ADS  Google Scholar 

  8. S. V. Manakov. Complete integrability and stochastization of discrete dynamic systems. Sov. Phys. JETP, 40(2):269–274, 1974.

    ADS  MathSciNet  Google Scholar 

  9. H. Flaschka. On the Toda lattice. II-inverse-scattering solution. Prog. Theor. Phys., 51(3):703–716, 1974.

    Article  ADS  MathSciNet  Google Scholar 

  10. J. Moser. Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math, 16(1), 1975.

    Google Scholar 

  11. J. Moser. Various aspects of integrable Hamiltonian systems. Dynamical Systems, CIME Lectures, Bressanone, Birkhäuser, Boston, 8, 1978.

    Google Scholar 

  12. Y. Kodama and J. Ye. Toda hierarchy with indefinite metric. Physica D, 91(4):321–339, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  13. S. P. Khastgir and R Sasaki. Instability of solitons in imaginary coupling affine Toda field theory. Prog. Theor. Phys., 95:485–501, 1996.

    Article  ADS  MathSciNet  Google Scholar 

  14. V. S. Gerdjikov, E. G. Evstatiev, and R. I. Ivanov. The complex Toda chains and the simple Lie algebras- solutions and large time asymptotics. J. Phys. A: Math. Gen., 31(40):8221–8232, 1998.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. V. S. Gerdjikov, B. B. Baizakov, and M. Salerno. Modeling adiabatic N-soliton interactions and perturbations. Theor. Math. Phys., 144(2):1138–1146, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  16. V. S. Gerdjikov, B. B. Baizakov, M. Salerno, and N. A. Kostov. Adiabatic N-soliton interactions of Bose–Einstein condensates in external potentials. Phys. Rev. E, 73(4):46606, 2006.

    Article  ADS  MathSciNet  Google Scholar 

  17. V. S. Gerdjikov and I. M. Uzunov. Adiabatic and non-adiabatic soliton interactions in nonlinear optics. Physica D: Nonl. Phen., 152:355–362, 2001.

    Article  ADS  MathSciNet  Google Scholar 

  18. V. S. Gerdjikov. On adiabatic N-soliton interactions and trace identities. Eur. Phys. J. B-Conden. Matter, 29(2):237–241, 2002.

    Article  ADS  MathSciNet  Google Scholar 

  19. V. S. Gerdjikov. Basic aspects of soliton theory. In Mladenov, I. M. and Hirshfeld, A. C., editor, Geometry, Integrability and Quantization, pages 78–125. Softex, Sofia, 2005.

    Google Scholar 

  20. V. S. Gerdjikov, E. V. Doktorov, and J. Yang. Adiabatic interaction of N ultrashort solitons: Universality of the complex Toda chain model. Phys. Rev. E, 64(5):56617, 2001.

    Article  ADS  MathSciNet  Google Scholar 

  21. V. S. Gerdjikov. N -soliton interactions, the complex Toda chain and stability of NLS soliton trains. In Prof. E. Kriezis, editor, Proceedings of the International Symposium on Electromagnetic Theory, volume 1 of Report Presented at the XVI-th International Symposium on Electromagnetic Theory URSI’98, Thessaloniki, Greece, May 25–28, pages 307–309. Aristotle University of Thessaloniki, Greece, 1998.

    Google Scholar 

  22. V. S. Gerdjikov, E. V. Doktorov, and N. P. Matsuka. N-soliton train and generalized complex Toda chain for the Manakov system. Theor. Math. Phys., 151(3):762–773, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  23. V. A. Marchenko. Sturm-Liouville Operators and Applications. Birkhäuser, Basel, 1987.

    Google Scholar 

  24. B. M. Levitan. Inverse Sturm-Liouville Problems. VSP Architecture, Zeist, 1987.

    MATH  Google Scholar 

  25. L. D. Faddeev. The inverse problem in the quantum theory of scattering. J. Math. Phys, 4(1):72–104, 1963.

    Article  ADS  MathSciNet  Google Scholar 

  26. F. Calogero, editor. Nonlinear Evolution Equations Solvable by the Spectral Transform, volume 26 of Res. Notes in Math. Pitman, London, 1978.

    Google Scholar 

  27. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. I. Pitaevskii. Theory of Solitons: The Inverse Scattering Method. Plenum, New York, 1984.

    Google Scholar 

  28. L. D. Faddeev and L. A. Takhtajan. Hamiltonian Methods in the Theory of Solitons. Springer-Verlag, Berlin, 1987.

    MATH  Google Scholar 

  29. M. J. Ablowitz and P. A. Clarkson. Solitons, Nonlinear Evolution Equations and Inverse Scattering, volume 149 of London Mathematical Society Lecture Notes Series. Cambridge University Press, Cambridge, 1991.

    Google Scholar 

  30. M. J. Ablowitz, A. D. Trubatch, and B. Prinari. Discrete and Continuous Nonlinear Schrodinger Systems. Cambridge University Press, Cambridge, 2003.

    Book  Google Scholar 

  31. I. D. Iliev, E. Kh. Christov, and K. P. Kirchev. Spectral Methods in Soliton Equations, volume 73 of Pitman Monographs and Surveys in Pure and Applied Mathematics. John Wiley & Sons, New York, 1991.

    Google Scholar 

  32. M. A. Naimark. Linear Differential Operators. Nauka, Moskow, 1969.

    Google Scholar 

  33. N. Dunford and J. T. Schwartz. Linear Operators. Part 1, 2, 3. Wiley Interscience Publications, New York 1971.

    Google Scholar 

  34. A. B. Shabat. Inverse-scattering problem for a system of differential equations. Funct. Anal. Its Appl., 9(3):244–247, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  35. A. B. Shabat. An inverse scattering problem. Diff. Equ., 15(10):1299–1307, 1979.

    MathSciNet  Google Scholar 

  36. A. V. Mikhailov. Reduction in integrable systems. The reduction group. JETP Lett., 32:174, 1980.

    ADS  Google Scholar 

  37. P. J. Caudrey. The inverse problem for the third order equation u xxx +q(x)u x +r(x)u = −iζ3 u. Phys. Lett. A, 79(4):264–268, 1980.

    Google Scholar 

  38. P. J. Caudrey. The inverse problem for a general N × N spectral equation. Physica D: Nonl. Phen., 6(1):51–66, 1982.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. R. Beals and R. R. Coifman. Scattering and inverse scattering for first order systems. Comm. Pure Appl. Math, 37:39–90, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  40. R. Beals and R. R. Coifman. Inverse scattering and evolution equations. Commun. Pure Appl. Math., 38(1):29–42, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  41. R. Beals and R. R. Coifman. The D-bar approach to inverse scattering and nonlinear evolutions. Physica D, 18(1–3):242–249, 1986.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  42. R. Beals and R. R. Coifman. Scattering and inverse scattering for first-order systems: II. Inverse Probl., 3(4):577–593, 1987.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  43. R. Beals and R. R. Coifman. Linear spectral problems, non-linear equations and the overline partial-method. Inverse Probl., 5(2):87–130, 1989.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  44. V. S. Gerdjikov and A. B. Yanovski. Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system. J. Math. Phys., 35:3687–3721, 1994.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  45. V. S. Gerdjikov. On the spectral theory of the Integro-ifferential operator generating nonlinear evolution equations. Lett. Math. Phys, 6:315–324, 1982.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  46. N. I. Muskhelischvili. Boundary Value Problems of Functions Theory and Their Applications to Mathematical Physics. Wolters-Noordhoff Publisher, Gröningen, The Netherlands, 1958.

    Google Scholar 

  47. F. D. Gakhov. Boundary Value Problems. Prtgamon Press, Oxford, 1966.

    Google Scholar 

  48. N. P. Vekua. Systems of Singular Integral Equations. P. Noordhoff Ltd, Gröningen, The Netherlands, 1967.

    MATH  Google Scholar 

  49. M. G. Gasimov. About an inverse problem for Sturm-Liouville equation. Dokl. Akad. Nauk. SSSR, 154(2), 1964.

    Google Scholar 

  50. V. E. Zakharov and S. V. Manakov. Exact theory of the resonans interaction of wave packets in nonlinear media. Preprint of the INF SOAN USSR 74-41, Novosibirsk, 1974.

    Google Scholar 

  51. V. E. Zakharov and A. B. Shabat. A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I. Funct. Anal. Appl., 8(3):226–235, 1974.

    Article  MATH  Google Scholar 

  52. V. E. Zakharov and A. B. Shabat. Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II. Funct. Anal. Appl., 13(3):166–174, 1979.

    MATH  MathSciNet  Google Scholar 

  53. V. E. Zakharov and A. V. Mikhailov. Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method. Sov. Phys. JETP, 47(6), 1978.

    Google Scholar 

  54. V. E. Zakharov and A. V. Mikhailov. On the integrability of classical spinor models in two-dimensional space-time. Commun. Math. Phys., 74(1):21–40, 1980.

    Article  ADS  MathSciNet  Google Scholar 

  55. A. V. Mikhailov. The reduction problem and the inverse scattering method. Physica D: Nonl. Phen., 3(1–2):73–117, 1981.

    Article  ADS  MATH  Google Scholar 

  56. V. S. Gerdjikov and G. G. Grahovski. Reductions and real forms of Hamiltonian systems related to N-wave type equations. Balkan Phys. Lett. BPL (Proc. Suppl.), BPU-4:531–534, 2000.

    Google Scholar 

  57. V. S. Gerdjikov, G. G. Grahovski, R. I. Ivanov, and N. A. Kostov. N-wave interactions related to simple Lie algebras. Inverse Probl., 17:999–1015, 2001.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  58. V. S. Gerdjikov, G. G. Grahovski, and N. A. Kostov. Reductions of N-wave interactions related to low-rank simple Lie algebras: I. Z_2-reductions. J. Phys. A: Math. Gen., 34(44):9425–9461, 2001.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  59. V. S. Gerdjikov, N. A. Kostov, and T. I. Valchev. N-wave equations with orthogonal algebras: Z_2 and Z_2× Z_2 reductions and soliton solutions. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 3, 2007.

    Google Scholar 

  60. D. J. Kaup and A. C. Newell. Solitons as particles, oscillators and in slowly varying media: A singular perturbation theory. Proc. R. Soc. Lond. A, 361:413, 1978.

    Article  ADS  Google Scholar 

  61. A. Bondeson, M. Lisak, and D. Anderson. Soliton perturbations: A variational principle for the parameters. Physica Scripta, 20:479–485, 1979.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  62. K. A. Gorshkov and L. A. Ostrovsky. Interactions of solitons in nonintegrable systems: Direct perturbation method and applications. Physica D: Nonl. Phen., 3(1–2):428–438, 1981.

    Article  ADS  MATH  Google Scholar 

  63. D. Anderson. Variational approach to nonlinear pulse propagation in optical fibers. Phys. Rev. A, 27(6):3135–3145, 1983.

    Article  ADS  Google Scholar 

  64. P. P. Kulish and V. N. Ed. Popov. Problems in Quantum Field Theory and Statistical Physics. Part V., volume 145 (in russian). Notes of LOMI Seminars, 1985.

    Google Scholar 

  65. R. J. Baxter. Exactly Solved Models in Statistical Mechanics. Academic Press, New York, 1982.

    MATH  Google Scholar 

  66. Y. Kodama and A. Hasegawa. Nonlinear pulse propagation in a monomode dielectric guide. IEEE J. Quantum Electron., 23(5):510–524, 1987.

    Article  ADS  Google Scholar 

  67. C. Desem. PhD thesis, University of New South Wales, Kensington, New South Wales, Australia, 1987.

    Google Scholar 

  68. D. J. Kaup. Perturbation theory for solitons in optical fibers. Phys. Rev. A, 42(9):5689–5694, 1990.

    Article  ADS  MathSciNet  Google Scholar 

  69. D. J. Kaup. Second-order perturbations for solitons in optical fibers. Phys. Rev. A, 44(7):4582–4590, 1991.

    Article  ADS  Google Scholar 

  70. L. Gagnon and P. A. Bélanger. Adiabatic amplification of optical solitons. Phys. Rev. A, 43(11):6187–6193, 1991.

    Article  ADS  Google Scholar 

  71. V. S. Gerdjikov and M. I. Ivanov. Expansions over the squared solutions and the inhomogeneous nonlinear Schrödinger equation. Inverse Probl., 8(6):831–847, 1992.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  72. N. A. Kostov and I. M. Uzunov. New kinds of periodical waves in birefringent optical fibers. Opt. Commun., 89(5–6):389–392, 1992.

    Article  ADS  Google Scholar 

  73. C. Desem and P. L. Chu. Soliton–Soliton Interaction. Optical Solitons-Theory and Experiment. Cambridge University Press, Cambridge, 1992.

    Google Scholar 

  74. J. M. Arnold. Soliton pulse-position modulation. IEEE Proc. J., 140(6):359–366, 1993.

    Google Scholar 

  75. I. M. Uzunov and V. S. Gerdjikov. Self-frequency shift of dark solitons in optical fibers. Phys. Rev. A, 47(2):1582–1585, 1993.

    Article  ADS  Google Scholar 

  76. I. M. Uzunov and V. S. Gerdjikov. Self-frequency shift of dark solitons in optical fibers. Phys. Rev. A, 47(2):1582–1585, 1993.

    Article  ADS  Google Scholar 

  77. J. M. Arnold. Stability theory for periodic pulse train solutions of the nonlinear Schrödinger equation. IMA J. Appl. Math., 50:123–140, 1994.

    Article  Google Scholar 

  78. J. M. Arnold. Stability of nonlinear pulse trains on optical fibers. Proceedings URSI Electromagnetic Theory Symposium, pages 553–555, St. Petersburg, 1995.

    Google Scholar 

  79. T. Okamawari, A. Hasegawa, and Y. Kodama. Analyses of soliton interactions by means of a perturbed inverse-scattering transform. Phys. Rev. A, 51(4):3203–3220, 1995.

    Article  ADS  Google Scholar 

  80. S. Wabnitz, Y. Kodama, and A. B. Aceves. Control of optical soliton interactions. Opt. Fiber Technol., 1:187–217, 1995.

    Article  ADS  Google Scholar 

  81. R. Radhakrishnan, A. Kundu, and M. Lakshmanan. Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: Integrability and soliton interaction in non-Kerr media. Phys. Rev. E, 60(3):3314–3323, 1999.

    Article  ADS  Google Scholar 

  82. V. S. Gerdjikov. Dynamical models of adiabatic N -soliton interactions. Balkan Phys. Lett. BPL (Proc. Suppl.), BPU-4:535–538, 2000.

    Google Scholar 

  83. V. S. Shchesnovich and J. Yang. Higher order solitons in N-wave system. Stud. Appl. Math., 110:297–332, 2005.

    Article  MathSciNet  Google Scholar 

  84. J. L. Lamb Jr. Analytical description of ultra-short optical pulse propagation in a resonant medium. Rev. Mod. Phys., 43:99–124, 1971.

    Article  ADS  MathSciNet  Google Scholar 

  85. 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.

    ADS  MathSciNet  Google Scholar 

  86. V. E. Zakharov and A. B. Shabat. Interaction between solitons in a stable medium. Sov. Phys. JETP, 37:823, 1973.

    ADS  Google Scholar 

  87. A. C. Scott, F. Y. F. Chu, and D. W. McLaughlin. The soliton: A new concept in applied science. Proc. IEEE, 61(10):1443–1483, 1973.

    Article  MathSciNet  ADS  Google Scholar 

  88. A. Hasegawa and F. Tappert. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Appl. Phys. Lett., 23:142–170, 1973.

    Article  ADS  Google Scholar 

  89. S. V. Manakov. Nonlinear Fraunhofer diffraction. Sov. Phys. JETP, 38:693, 1974.

    ADS  MathSciNet  Google Scholar 

  90. J. Satsuma and N. Yajima. Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media. Prog. Theor. Phys. Suppl., 55:284, 1974.

    Article  ADS  MathSciNet  Google Scholar 

  91. V. E. Zakharov and S. V. Manakov. On the complete integrability of a nonlinear Schrödinger equation. Theoreticheskaya i Mathematicheskaya Fizika, 19(3):332–343, 1974.

    MATH  Google Scholar 

  92. L. A. Takhtadjan. Exact theory of propagation of ultrashort optical pulses in two-level media. J. Exp. Theor. Phys., 39(2):228–233, 1974.

    ADS  Google Scholar 

  93. S. V. Manakov. On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Sov. Phys. JETP, 38:248–253, 1974.

    ADS  MathSciNet  Google Scholar 

  94. V. E. Zakharov and S. V. Manakov. The theory of resonant interactions of wave packets in nonlinear media. Zh. Eksp. Teor. Fiz, 69(5), 1975.

    Google Scholar 

  95. DJ Kaup, A. Reiman, and A. Bers. Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium. Rev. Mod. Phys., 51(2):275–309, 1979.

    Article  ADS  MathSciNet  Google Scholar 

  96. C. Cercignani. Solitons-theory and application. Nuovo Cimento, Rivista, Serie, 7:429–469, 1977.

    Article  ADS  MathSciNet  Google Scholar 

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

    Google Scholar 

  98. S. J. Orfanidis. Discrete sine-Gordon equations. Phys. Rev. D, 18(10):3822–3827, 1978.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  99. S. J. Orfanidis. Sine-Gordon equation and nonlinear σ model on a lattice. Phys. Rev. D, 18(10):3828–3832, 1978.

    Article  ADS  MathSciNet  Google Scholar 

  100. K. Longren and A. Ed. Scott. Solitons in Action. Academic Press, New York, 1978.

    Google Scholar 

  101. D. J. Kaup, A. Reiman, and A. Bers. Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium. Rev. Mod. Phys., 51(2):275–309, 1979.

    Article  ADS  MathSciNet  Google Scholar 

  102. R. K. Bullough and P. J. Caudrey, editors. Solitons. Springer, Berlin, 1980.

    MATH  Google Scholar 

  103. G. Eilemberger. Solitons, volume 9 of Mathematical Methods for Scientists. Solid State Sciences. Springer-Verlag, Berlin, 1981.

    Google Scholar 

  104. S. V. Manakov and V. E. Zakharov. Three-dimensional model of relativistic-invariant field theory, integrable by the inverse scattering transform. Lett. Math. Phys., 5(3):247–253, 1981.

    Article  ADS  MathSciNet  Google Scholar 

  105. F. Calogero and A. Degasperis. Spectral Transform and Solitons. I. Tools to Solve and Investigate Nonlinear Evolution Equations, volume 144 of Studies in Mathematics and Its Applications, 13. Lecture Notes in Computer Science. North-Holland Publishing Co., Amsterdam, New York, 1982.

    Google Scholar 

  106. W. Oevel. On the integrability of the Hirota-Satsuma system. Phys. Lett. A, 94(9):404–407, 1983.

    Article  ADS  MathSciNet  Google Scholar 

  107. J. L. Lamb Jr. Elements of Soliton Theory. Wiley, New York, 1980.

    MATH  Google Scholar 

  108. J. J-P. Leon. Integrable sine-Gordon model involving external arbitrary field. Phys. Rev. A, 30(5):2830–2836, 1984.

    Article  ADS  Google Scholar 

  109. Shastry, B. S., Jha, S. S., and Singh, V. (eds).: Exactly Solvable Problems in Condensed Matter and Relativistic Field Theory, Lect. Notes Phys. 242. Springer Verlag, Berlin (1985)

    Google Scholar 

  110. A. C. Newell. Solitons in Mathematics and Physics. Regional Conf. Ser. in Appl. Math. Philadelphia, 1985.

    Google Scholar 

  111. K. B. Wolf. Symmetry in Lie optics. Ann. Phys., 172(1):1–25, 1986.

    Article  MATH  ADS  Google Scholar 

  112. Y. S. Kivshar and B. A. Malomed. Dynamics of solitons in nearly integrable systems. Rev. Mod. Phys., 61(4):763–915, 1989.

    Article  ADS  Google Scholar 

  113. P. G. Drazin and R. S. Johnson. Solitons: An Introduction. Cambridge texts in Applied Mathematics. Cambridge University Press, Cambridge, 1989.

    Google Scholar 

  114. E. E. Infeld and G. Rowlands. Nonlinear Waves, Solitons and Chaos. Cambridge University Press, Cambridge, 1990.

    MATH  Google Scholar 

  115. V. E. Vekslerchik and V. V. Konotop. Discrete nonlinear Schrödinger equation under nonvanishing boundary conditions. Inverse Probl., 8(6):889–909, 1992.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  116. V. E. Zakharov, editor. What is Integrability? Springer series in Nonlinear Dynamics. Springer Verlag, Berlin, 1992.

    Google Scholar 

  117. A. C. Scott. Davydovs soliton. Phys. Rep., 217(1):1–67, 1992.

    Article  ADS  Google Scholar 

  118. G. R. Agrawal. Nonlinear Fiber Optics. Elsevier, Amsterdam, 2001.

    Google Scholar 

  119. A Hasegawa and Y Kodama. Solitons in Optical Communications. Oxford University Press, New York, 1995.

    MATH  Google Scholar 

  120. S. Kakei, N. Sasa, and J. Satsuma. Bilinearization of a generalized derivative nonlinear Schrödinger equation. J. Phys. Soc. Japan, 64(5):1519–1523, 1995.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  121. A. A. Sukhorukov and N. N. Akhmediev. Multisoliton complexes on a background. Phys. Rev. E, 61(5):5893–5899, 2000.

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Gerdjikov, V., Vilasi, G., Yanovski, A. (2008). The Inverse Scattering Problem for the Zakharov–Shabat System. In: Gerdjikov, V., Vilasi, G., Yanovski, A. (eds) Integrable Hamiltonian Hierarchies. Lecture Notes in Physics, vol 748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77054-1_4

Download citation

Publish with us

Policies and ethics