On the stability of solitary waves for classical scalar fields

Ph. Blanchard; J. Stubbe; L. Vázquez

Annales de l'I.H.P. Physique théorique (1987)

  • Volume: 47, Issue: 3, page 309-336
  • ISSN: 0246-0211

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Blanchard, Ph., Stubbe, J., and Vázquez, L.. "On the stability of solitary waves for classical scalar fields." Annales de l'I.H.P. Physique théorique 47.3 (1987): 309-336. <http://eudml.org/doc/76381>.

@article{Blanchard1987,
author = {Blanchard, Ph., Stubbe, J., Vázquez, L.},
journal = {Annales de l'I.H.P. Physique théorique},
keywords = {stability; bound states of lowest action; nonlinear Klein-Gordon and Schrödinger equations; Shatah-Strauss formalism; existence; minimum action solutions; logarithmic Klein-Gordon; logarithmic Schrödinger; scalar field equations; fractional nonlinearitities},
language = {eng},
number = {3},
pages = {309-336},
publisher = {Gauthier-Villars},
title = {On the stability of solitary waves for classical scalar fields},
url = {http://eudml.org/doc/76381},
volume = {47},
year = {1987},
}

TY - JOUR
AU - Blanchard, Ph.
AU - Stubbe, J.
AU - Vázquez, L.
TI - On the stability of solitary waves for classical scalar fields
JO - Annales de l'I.H.P. Physique théorique
PY - 1987
PB - Gauthier-Villars
VL - 47
IS - 3
SP - 309
EP - 336
LA - eng
KW - stability; bound states of lowest action; nonlinear Klein-Gordon and Schrödinger equations; Shatah-Strauss formalism; existence; minimum action solutions; logarithmic Klein-Gordon; logarithmic Schrödinger; scalar field equations; fractional nonlinearitities
UR - http://eudml.org/doc/76381
ER -

References

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  1. [1] J. Stubbe, Constrained minimization problems in Orlicz-spaces with application to minimum action solutions of non-linear scalar field equations in RN. Bielefeld University, BI-TP 86/10, 1986. 
  2. [2] J. Shatah, Stable Standing waves of Nonlinear Klein-Gordon equations. Comm. Math. Phys., t. 91, 1983, p. 313-327. Zbl0539.35067MR723756
  3. [3] J. Shatah and W. Strauss, Instability of nonlinear bound-States. Comm. Math. Phys., t. 100, 1985, p. 173-190. Zbl0603.35007MR804458
  4. [4] W. Strauss, Existence of solitary waves in higher dimensions. Comm. Math. Phys., t. 55, 1977, p. 149-162. Zbl0356.35028MR454365
  5. [5] H. Berestycki and P.L. Lions, Nonlinear Scalar Field Equations. I. Arch. Rat. Mech. Anal., t. 82, 1983, p. 313-345. Zbl0533.35029MR695535
  6. [6] H. Brezis and E.H. Lieb, Minimum Action Solution of some Vector field equation. Comm. Math. Phys., t. 96, 1984, p. 97-113. Zbl0579.35025MR765961
  7. [7] H. Pecher, Low energy scattering for nonlinear Klein-Gordon equations, preprint. MR795519
  8. [8] W. Strauss, On weak solutions of semilinear hyperbolic equations. An. Acad. brasil. Cienc., t. 42, 1970, p. 645-651. Zbl0217.13104MR306715
  9. [9] J. Ginibre and G. Velo, Non-linear evolution equations: Cauchy problem and scattering theory, BIBOS preprint 102/85, Bielefeld, 1985. 
  10. [10] J. Ginibre and G. Velo, On a class of Nonlinear Schrödinger equations. I. The Cauchy problem, General case. J. Funct. Anal., t. 32, 1979, p. 1-32. Zbl0396.35028MR533218
  11. [11] T. Cazenave and A. Haraux, Équations d'évolution avec non linéarité logarithmique. Aunals. Fac. Sci. Univ. Toulouse, t. 2, 1980, p. 21-55. Zbl0411.35051MR583902
  12. [12] A. Haraux, Nonlinear Evolution Equations. Global Behaviour of Solutions. Lecture Notes in Mathematics, t. 841, Springer, Berlin, 1981. Zbl0461.35002MR610796
  13. [13] T.F. Morris, Classical confinement: field theories with spontaneously bounded domains. Had. J., t. 3, 1980, p. 1333-1359. Zbl0444.49038MR588061
  14. [14] G. Rosen, Dilation Covariance and Exact Solutions in Local Relativistic Field Theories. Phys. Rev., t. 183, 1969, p. 1186-1188. 
  15. [15] I. Bialynicki-Birula and J. Michielsky, Wave Equations with Logarithmic Non–linearities. Bull. Acad. Pol. Sci. Cl. II., t. 23, 1975, p. 461-466. MR403458
  16. [16] T.F. Morris, Classical Theory of Klein-Gordon equations with logarithmic non–linearities. Can. Journ. Phys., t. 56, 1978, p. 1405-1411. Zbl1043.81526MR509820
  17. [17] I. Bialynicki-Birula and J. Michielsky, Nonlinear Wave Mechanics, Ann. of Phys., t. 100, 1976, p. 62-93. MR426670
  18. [18] G. Marques and I. Ventura, Resonances within nonperturbative methods in field theories. Phys. Rev., t. D 14, 1975, p. 1056-1059. 
  19. [19] M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys., t. 87, 1983, p. 567-576. Zbl0527.35023MR691044
  20. [20] H. Berestycki and Th. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon nonlinéaires. C. R. Acad. Sc. Paris, t. 293, 1981, p. 489-492. Zbl0492.35010MR646873
  21. [21] T. Cazenave and P.L. Lions, Orbital Stability of Standing waves for some Nonlinear Schrödinger Equations. Comm. Math. Phys., t. 85, 1982, p. 549-561. Zbl0513.35007MR677997
  22. [22] M. Weinstein, Lyapunov Stability of Ground States of Nonlinear Dispersive Evolution equations. Comm. Pure Appl. Math., t. 39, 1986, p. 51-67. Zbl0594.35005MR820338
  23. [23] T. Cazenave, Stable solutions of the logarithmic Schrödinger equation. Nonlinear Analysis TMA, t. 7, 1983, p. 1127-1140. Zbl0529.35068MR719365
  24. [24] E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Stud. Appl. Math., t. 57, 1977, p. 93-105. Zbl0369.35022MR471785
  25. [25] G. Ginibre and G. Velo, On a class of nonlinear Schrödinger equation with nonlocal interaction. Math. Zeitschrift, t. 170, 1980, p. 109-136. Zbl0407.35063MR562582
  26. [26] Ph Blanchard, J. Stubbe and L. Vázquez, On the stability of nonlinear spinor fields, Phys. Rev. D, to appear. Zbl0649.35076
  27. [27] G.H. Derrick, Comments on nonlinear wave equations as models for elementary particles. J. Math. Phys., t. 5, 1964, p. 1252-1254. MR174304
  28. [28] M. Weinstein, Modulational stability of Ground states of nonlinear Schrödinger Equations. Siam J. Math. Anal., t. 16, 1985, p. 472-491. Zbl0583.35028MR783974
  29. [29] M. Grillakis, J. Shatah and W. Strauss, Stability Theory of Solitary Waves in the Presence of Symmetry, I. J. Funct. Anal., t. 74, 1987, p. 160-197. Zbl0656.35122MR901236

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