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Soliton Stability

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Soliton Phenomenology

Part of the book series: Mathematics and Its Applications () ((MASS,volume 33))

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Abstract

We first define the stability of soliton-like solutions. As has been mentioned above, these describe extremum states of some nonlinear system. There are two types of system stability: (1) with respect to a perturbation of the initial data, and (2) with respect to a perturbation of the evolution equation which describes the system behaviour (the structural stability).

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References

  1. R.K. Dodd, et al. see [34] part I.

    Google Scholar 

  2. Ph.M. Morse and H. Feshbach. Methods of theoretical physics. McGraw-Hill N.Y., London 1953, V, I and II.

    MATH  Google Scholar 

  3. I.M. Krichever. On rational solutions of KP equation and integrable N-particle systems on line. Funk. Anal. Priloz. 1978, 12, p. 76–78.

    MATH  Google Scholar 

  4. I.M. Krichever. On rational solutions to Zakharov-Shabat equations for complete integrable N-particle systems on line. Zap. Nauch. Semin. LOMI, 1979, 84, p. 117–130.

    MATH  MathSciNet  Google Scholar 

  5. I.V. Cherednik. Funk. Anal. Priloz. 1978, 12, p. 45–52.

    MATH  MathSciNet  Google Scholar 

  6. V.M. Eleonskii, I.M. Krichever and I.M. Kulagin. Rational multi-soliton solutions of NLS. Doklady AN 1986, 287, p. 606–610.

    MathSciNet  Google Scholar 

  7. I.M. Krichever. Spectral theory of ‘finite-band’ time-dependent Schrödinger operators. Non-stationary Peierls model. Funk. Anal. Priloz. 1986, 20, p. 42–54. See [89] part I.

    MathSciNet  Google Scholar 

  8. V. Makhankov, R. Myrzakulov and Yu.V. Katyshev. Vector generalization of the system of equations for interacting h.f. and If waves. Preprint JINR PI 7-86-94, Dubna 1986. Teor. Mat. Fiz. 1987, 72, p. 22–34.

    Google Scholar 

  9. V.G. Makhankov. On stationary solutions of Schrödinger equation with the self consistent potential satisfying the Boussinesq equation. Phys. Lett. 1974, 50A, p. 42–44.

    ADS  Google Scholar 

  10. V. Makhankov. Stationary solutions of coupled Schrödinger and Boussinesq equations and dynamics of Langmuir waves. Preprint JINR E5-8390, Dubna 1974.

    Google Scholar 

  11. K. Nishikawa, et al. see [39] part I.

    Google Scholar 

  12. Ya. Bogomolov, I.A. Kolchugina, A.G. Litvak and A.M. Sergeev. Near-sonic Langmuir solitons. Phys. Lett. 1982, 91A, p. 447–450.

    ADS  Google Scholar 

  13. H. Kramer, E.W. Laedke and K.H. Spatschek. Transition from standing to sonic Langmuir solitons. Phys. Rev. Lett. 1984, 52, p. 1226–1229.

    Article  ADS  Google Scholar 

  14. V. Makhankov. See [7] part I.

    Google Scholar 

  15. K. Mima, K. Kato and K. Nishikawa. Self-consistent stationary density cavities with many hounded Langmuir waves. J. Phys. Soc. Jpn. 1977, 42, p. 290–296.

    Article  ADS  Google Scholar 

  16. H. Hojo and K. Nishikawa. On bounded Langmuir waves in self-consistent density cavity. J. Phys. Soc. Jpn. 1977, 42, p. 1437–1438.

    Article  ADS  Google Scholar 

  17. I.M. Krichever, et al. see [17] part III.

    Google Scholar 

  18. See, e.g. A.C. Newell. The inverse scattering transform, on R. Bullough and P. Caudrey eds. Solitons. Springer, Heidelberg 1980, p. 177–242, formula (6.170) p. 215.

    Google Scholar 

  19. R. Finkelstein, R. Levelier and M. Ruderman. Phys. Rev. 1951, 83, p. 326-. R. Finkelstein, C. Fronsdal and P. Kaus, ibid, 1956, 103, p. 1571-.

    Article  MATH  ADS  Google Scholar 

  20. V.G. Makhankov. On the existence of non-one-dimensional soliton-like solutions for some field theories. Phys. Lett. 1977, 61A, p. 431–433.

    ADS  Google Scholar 

  21. I.V. Amirkhanov and E.P. Zhidkov. Existence of positively-defined particlelike solution of nonlinear equation of φ4–φ6 theory. JINR P5-82-246, Dubna 1982.

    Google Scholar 

  22. V.G. Makhankov and P.E. Zhidkov. On the existence of static hole-like solutions in a nonlinear Bose gas model. Comm. of JINR P5-86-341, Dubna 1986.

    Google Scholar 

  23. R. Friedberg, T.D. Lee and A. Sirlin. Class of scalar-field soliton solution in three space-dimensions. Phys. Rev. 1976, 12D, p. 2739–2761.

    ADS  MathSciNet  Google Scholar 

  24. L. Nehari. Proc. Roy. Irish Acad. 1963, 62A, p. 118-.

    MathSciNet  Google Scholar 

  25. E.P. Shidkov and V.P. Shirikov. On a boundary problem for the second order ordinary differential equation. Z. Vych. Mat. Mat. Fiz. 1964, 4, p. 804-.

    Google Scholar 

  26. A. Scott, F. Chu and D. McLaughlin. The soliton: A new concept in applied science. Proc. IEEE 1973, 61, p. 1443–1483.

    Article  ADS  MathSciNet  Google Scholar 

  27. E. Laedke and K.H. Spatschek. Nonlinear stability of envelope solitons. Phys. Rev. Lett. 1978, 41, p. 1798–1801. Exact stability criteria for finite-amplitude solitons. Phys. Rev. Lett. 1972, 42, p. 1534–1537.

    Article  ADS  Google Scholar 

  28. L.D. Faddeev. In search for many-dimensional solitons, in Proc. Non-local, Non-linear and Non-renormalized Field Theories, Alushta 1976. JINR publishing D2-9788, Dubna 1976. p. 207–223. A.S. Schvartz. Topologically non-trivial solutions of classical equations and their role in QFT. ibid p. 224–240.

    Google Scholar 

  29. E. Ott and R. Sudan. Nonlinear theory of ion-acoustic waves with Landau damping. Phys. Fluids 1969, 12, p. 3288–3294.

    Article  MathSciNet  Google Scholar 

  30. J. Keener and D. McLaughlin. Solitons under perturbations. Phys. Rev. 1977, 16A, p. 777–790.

    ADS  MathSciNet  Google Scholar 

  31. M. Fogel, S. Trullinger, A. Bishop and J. Krumhansl. Classical particle-like behavior of SG solitons in scattering potentials and applied fields. Phys. Rev. Lett. 1976, 36, p. 1411–1414.

    Article  ADS  MathSciNet  Google Scholar 

  32. Y. Ichikawa. Topics on solitons in plasmas. Phys. Scripta 1979, 20, p. 296–305.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  33. V.I. Karpman. Soliton evolution in the presence of perturbation, ibid. p. 462–478 and references therein.

    Google Scholar 

  34. A. Bondeson, M. Lisak and D. Anderson. Soliton perturbations. A variational principle for the soliton parameters, ibid, p. 479–485 and Stability analysis of lower-hybrid cones, p. 343–345.

    Google Scholar 

  35. V.K. Fedyanin and V. Makhankov. Soliton-like solutions in one-dimensional systems with resonance interaction. Phys. Scripta 1979, 20, p. 552–557.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  36. V Arnold. Sov. Math. Surveys 1963, 18, p. 9–36. J. Moser. Nearly integrable and integrable systems. Preprint N.Y. 1976 and Lectures on celestial mechanics. Springer, N.Y. 1971 (with C. Siegel).

    ADS  Google Scholar 

  37. V. Zakharov and A. Shabat. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. See [37] part II.

    Google Scholar 

  38. V. Makhankov, O.K. Pashaev and S. Sergeenkov. Colour solutions in a stable medium. See [15] part III.

    Google Scholar 

  39. J. Berryman. Stability of solitary waves in shallow water. Phys. Fluids 1976, 19, p. 771–777.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  40. J. Satsuma and N. Yajima. Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media. Suppl. Progr. Theor. Phys. 1974, 55, p. 284–306.

    Article  ADS  MathSciNet  Google Scholar 

  41. L.G Zastavenko. Particle-like solutions of nonlinear wave equation. Prik. Mat. Mekh. 1965, 29 p. 430–439.

    MathSciNet  Google Scholar 

  42. I.V. Barashenkov, A.D. Gocheva, V. Makhankov and I.V. Pusynin. On stability of solitary waves of the φ4–φ6 NLS under non-vanishing boundary conditions. JINR E17-85-967, Dubna 1985. See [41] part III.

    Google Scholar 

  43. T.B. Benjamin. Stability of solitary waves. Proc. Roy. Soc. Lond. 1972, 238, p. 153–183.

    ADS  MathSciNet  Google Scholar 

  44. D. Anderson. Stability of time-dependent particle-like solutions in nonlinear field theories. II. J. Math. Phys. 1971, 12, p. 945–952. D. Anderson and G. Derrick. Stability of time-dependent... part I. ibid, 1970, 11, p. 1336–1346.

    Article  ADS  Google Scholar 

  45. V.G. Makhankov. On stability of ‘charged’ solitons in the framework of Klein-Gordon equation with a saturable nonlinearity. JINR Commun. P2-10362, Dubna, 1977.

    Google Scholar 

  46. R. Friedberg, T.D. Lee and A. Sirlin. Class of field soliton solutions in 3-D space. Phys. Rev. 1976, 13D, p. 2739–2761. See [23] this part.

    ADS  MathSciNet  Google Scholar 

  47. Yu.P. Rybakov. Conditional stability of regular solutions of nonlinear field theory. In Problems of Gravity and Particles Theory. Atomizdat, Moscow 1979, p. 194–202.

    Google Scholar 

  48. N.G. Vakhitov and A.A. Kolokolov. See [53b] part I.

    Google Scholar 

  49. Yu.P. Rybakov. Stability of solitons. In Problems of Gravity and Particles Theory, issue 16. Energoatomizdat, Moscow 1985.

    Google Scholar 

  50. T. Yoshizawa. On the stability of solutions of a system of the differential equations. Mem. Coll. Sci. Univ. Kyoto Math. ser. 1955, 24A, No. 1. A.M. SLobodkin. On stability of consetyative system equilibrium with infinite number of degrees of freedom. Prikl. Mat. Mekh. 1962, 26, p. 356–358.

    Google Scholar 

  51. T. Cazenave and P.L. Lions. Orbital stability of standing waves for some NLS. Comm. Math. Phys. 1982, 85, p. 549–561.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  52. A.A. Movchan. Stability of processes with respect two metrics. Pricl. Mat. Mekh. 1960, 24, p. 988–1001.

    Google Scholar 

  53. Yu.P. Rybakov. On stability of multi-charged solitons. In Problems of Gravity and Particles Theory, issue 14. Energoizdat Moscow, 1983.

    Google Scholar 

  54. L. Shvartz. Analyse Mathématique. Hermann, Paris 1967. Vol. I, ch. III.

    Google Scholar 

  55. I.V. Barashenkov. Stability of solutions to nonlinear models possessing a sign-undefined metric. Acta Phys. Austriaca 1983, 55, p. 155–165.

    MathSciNet  Google Scholar 

  56. R. Jackiv and P. Rossi. Stability and bifurcation in Yang-Mills theory. Phys. Rev. 1980, 21D, p. 426–445.

    ADS  Google Scholar 

  57. A. Kummar, V. Nisichenko and Yu. Rybakov. Stability of charged solitons. Int. J. of Ther. Phys. 1979, 18, p. 425–432.

    Article  Google Scholar 

  58. V. Katyshev, N. Makhaldiani and V. Makhankov. On stability of soliton solutions to the NLS with nonlinearity ∣ψ∣νψ. Phys. Lett. 1978, 66A, p. 456–458.

    ADS  MathSciNet  Google Scholar 

  59. I.V. Barashenkov. Stability of ‘coloured’ solitons. JINR Comm. P2-82-376, Dubna 1982.

    Google Scholar 

  60. E. Ott and R. Sudan. See [29a] this part.

    Google Scholar 

  61. A. Hasegawa and F. Tappert. Transition of stationary nonlinear optical pulses in dispersive dielectric fibres. I and II. Appl. Phys. Lett. 1973, 23, p. 142–144 and p. 171–172.

    Article  ADS  Google Scholar 

  62. N. Yajima, M. Oikawa, J. Satsuma and C. Namba. Res. Inst. Appl. Phys. Repts. XXII, 1975, 70, p. 89.

    Google Scholar 

  63. N. Pereira. Soliton in the dampled NLS. Phys. Fluids 1977, 20, p. 1735–1743.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  64. J. Fernandes and G. Reinish. See [88] part I.

    Google Scholar 

  65. J. Fernandes, G. Reinish, A. Bondeson and J. Weiland. Collapse of a KdV soliton into a weak noise shelf. Phys. Lett. 1978, 66A, p. 175–178.

    ADS  Google Scholar 

  66. S. Watanabe. Soliton and generation of tail in nonlinear dispersive media with weak dissipation. J. Phys. Soc. Jpn. 1978, 45, p. 276–282.

    Article  ADS  Google Scholar 

  67. B. Cohen, K. Watson and B.J. West. Some properties of deep water solitons. Phys. Fluids 1976, 19, p. 345–354.

    Article  MATH  ADS  Google Scholar 

  68. B.A. Dubrovin et al. See [17] part III, Sov. J. Particles & Nuclei. 1988, 19, p. 252–269.

    MathSciNet  Google Scholar 

  69. V.G. Makhankov and Kh.T. Kholmurodov. Numerical study of stability of vector U(2) solitons. JINR Rapid Communications. No. 5 [25]-87. Dubna, 1987.

    Google Scholar 

  70. I.V. Barashenkov and Kh. Kholmurodov. Bose-gas with two-and three-body interactions: evolution of the unstable ‘bubbles’. JINR Communication, P17-86-698, Dubna, 1986 (in Russian).

    Google Scholar 

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  1. Laboratory of Computing Techniques and Automation, Joint Institute for Nuclear Research, Moscow, U.S.S.R.

    Vladimir G. Makhankov

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  1. Vladimir G. Makhankov
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© 1990 Kluwer Academic Publishers

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Makhankov, V.G. (1990). Soliton Stability. In: Soliton Phenomenology. Mathematics and Its Applications (Soviet Series), vol 33. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2217-4_11

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  • DOI: https://doi.org/10.1007/978-94-009-2217-4_11

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-7494-0

  • Online ISBN: 978-94-009-2217-4

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