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Functional Analysis and Its Applications

, Volume 48, Issue 1, pp 24–35 | Cite as

Absence of solitons with sufficient algebraic localization for the Novikov-Veselov equation at nonzero energy

  • A. V. KazeykinaEmail author
Article

Abstract

It is shown that the Novikov-Veselov equation (an analogue of the KdV equation in dimension 2 + 1) at positive and negative energies does not have solitons with space localization stronger than O(|x|−3) as |x| →∞.

Key words

traveling wave localized soliton Novikov-Veselov equation 

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References

  1. [1]
    M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press, Cambridge, 1991.CrossRefzbMATHGoogle Scholar
  2. [2]
    M. Boiti, J.J.-P. Leon, M. Manna, and F. Pempinelli, “On a spectral transform of a KdVlike equation related to the Schrödinger operator in the plane,” Inverse Problems, 3:1 (1987), 25–36.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    M. Boiti, J.J.-P. Leon, L. Martina, and F. Pempinelli, “Scattering of localized solitons in the plane,” Phys. Lett. A., 132 (1988), 432–439.CrossRefMathSciNetGoogle Scholar
  4. [4]
    A. de Bouard and J.-C. Saut, “Solitary waves of generalized Kadomtsev-Petviashvili equations,” Ann. Inst. H. Poincaré, Analyse Non Linéaire, 14:2 (1997), 211–236.CrossRefzbMATHGoogle Scholar
  5. [5]
    A. de Bouard and J.-C. Saut, “Symmetries and decay of the generalized Kadomtsev-Petviashvili solitary waves,” SIAM J. Math. Anal., 28:5 (1997), 1064–1085.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    L. D. Faddeev, “Increasing solutions of the Schrödinger equation,” Dokl. Akad. Nauk SSSR, 165:3 (1965), 514–517; English transl.: Soviet Phys. Dokl., 10 (1966), 1033–1035.MathSciNetGoogle Scholar
  7. [7]
    L. D. Faddeev, “The inverse problem in the quantum theory of scattering. II,” in: Itogi Nauki i Tekhniki. Sovremennye Problemy Matematiki, vol. 3, VINITI, Moscow, 1974, 93–180; English transl.: J. Soviet Math., 5:3 (1976), 334–396.Google Scholar
  8. [8]
    A. S. Fokas and M. J. Ablowitz, “On the inverse scattering of the time-dependent Schrödinger equation and the associated Kadomtsev-Petviashvili equation,” Stud. Appl. Math., 69:3 (1983), 211–228.zbMATHMathSciNetGoogle Scholar
  9. [9]
    A. S. Fokas and P. M. Santini, “Coherent structures in multidimensions,” Phys. Rev. Lett., 63:13 (1983), 1329–1333.CrossRefMathSciNetGoogle Scholar
  10. [10]
    I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Amer. Math. Soc., Providence, RI, 1969.zbMATHGoogle Scholar
  11. [11]
    P. G. Grinevich, “Rational solitons of the Veselov-Novikov equation are reflectionless potentials at fixed energy,” Teoret. Mat. Fiz., 69:2 (1986), 307–310; English transl.: Theoret. Math. Phys., 69 (1986), 1170–1172.MathSciNetGoogle Scholar
  12. [12]
    P. G. Grinevich, “Scattering transformation at fixed non-zero energy for the two-dimensional Schrödinger operator with potential decaying at infinity,” Uspekhi Mat. Nauk, 55:6 (2000), 3–70; English transl.: Russian Math. Surveys, 55:6 (2000), 1015–1083.CrossRefMathSciNetGoogle Scholar
  13. [13]
    P. G. Grinevich and R. G. Novikov, “Analogues of multisoliton potentials for the two-dimensional Schrödinger operator, and a nonlocal Riemann problem,” Dokl. Akad. Nauk SSSR, 286:1 (1986), 19–22; English transl.: Soviet Math. Dokl., 33:1 (1986), 9–12.MathSciNetGoogle Scholar
  14. [14]
    P. G. Grinevich and S. P. Novikov, “Two-dimensional’ inverse scattering problem’ for negative energies and generalized-analytic functions. I. Energies below the ground state,” Funkts. Anal. Prilozhen., 22:1 (1988), 23–33; English transl.: Functional Anal. Appl., 22:1 (1988), 19–27.MathSciNetGoogle Scholar
  15. [15]
    R. G. Novikov and G. M. Khenkin, “The \(\bar \partial \)-equation in the multidimensional inverse scattering problem,” Uspekhi Mat. Nauk, 42:3 (1987), 93–152; English transl.: Russian Math. Surveys, 42:3 (1987), 109–180.MathSciNetGoogle Scholar
  16. [16]
    A. V. Kazeykina, “A large time asymptotics for the solution of the Cauchy problem for the Novikov-Veselov equation at negative energy with non-singular scattering data,” Inverse Problems, 28:5 (2012), 055017.CrossRefMathSciNetGoogle Scholar
  17. [17]
    A. V. Kazeykina, “Absence of conductivity-type solutions for the Novikov-Veselov equation at zero energy,” Funkts. Anal. Prilozhen., 47:1 (2013), 79–82; English transl.: Functional Anal. Appl., 47:1 (2013), 64–66.CrossRefGoogle Scholar
  18. [18]
    A. V. Kazeykina, Absence of solitons with sufficient algebraic localization for the Novikov-Veselov equation at nonzero energy, http://arxiv.org/abs/1201.2758.
  19. [19]
    A. V. Kazeykina and R. G. Novikov, “A large time asymptotics for transparent potentials for the Novikov-Veselov equation at positive energy,” J. Nonlinear Math. Phys., 18:3 (2011), 377–400.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [20]
    A. V. Kazeykina and R. G. Novikov, “Large time asymptotics for the Grinevich-Zakharov potentials,” Bull. Sci. Math., 135:4 (2011), 374–382.CrossRefzbMATHMathSciNetGoogle Scholar
  21. [21]
    A. V. Kazeykina and R. G. Novikov, “Absence of exponentially localized solitons for the Novikov-Veselov equation at negative energy,” Nonlinearity, 24:6 (2011), 1821–1830.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [22]
    S. V. Manakov, “The inverse scattering method and two-dimensional evolution equations,” Uspekhi Mat. Nauk, 31:5 (1976), 245–246.zbMATHMathSciNetGoogle Scholar
  23. [23]
    R. G. Novikov, “The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator,” J. Funct. Anal., 103:2 (1992), 409–463.CrossRefzbMATHMathSciNetGoogle Scholar
  24. [24]
    R. G. Novikov, “Absence of exponentially localized solitons for the Novikov-Veselov equation at positive energy,” Phys. Letters A, 375:9 (2011), 1233–1235.CrossRefzbMATHGoogle Scholar
  25. [25]
    A. P. Veselov and S. P. Novikov, “Finite-zone, two-dimensional, potential Schrödinger operators. Explicit formulas and evolutions equations,” Dokl. Akad. Nauk SSSR, 279 (1984), 20–24; English transl.: Soviet Math. Dokl., 30 (1984), 588–591.MathSciNetGoogle Scholar
  26. [26]
    A. P. Veselov and S. P. Novikov, “Finite-zone, two-dimensional Schrödinger operators. Potential operators,” Dokl. Akad. Nauk SSSR, 279 (1984), 784–788; English transl.: Soviet Math. Dokl., 30 (1984), 705–708.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Ecole PolytechniqueMoscow State University CMAPMoscowFrance

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