On the ground states of vector nonlinear Schrödinger equations

Thierry Colin; Michael I. Weinstein

Annales de l'I.H.P. Physique théorique (1996)

  • Volume: 65, Issue: 1, page 57-79
  • ISSN: 0246-0211

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Colin, Thierry, and Weinstein, Michael I.. "On the ground states of vector nonlinear Schrödinger equations." Annales de l'I.H.P. Physique théorique 65.1 (1996): 57-79. <http://eudml.org/doc/76736>.

@article{Colin1996,
author = {Colin, Thierry, Weinstein, Michael I.},
journal = {Annales de l'I.H.P. Physique théorique},
keywords = {ground states; vector nonlinear Schrödinger equation; limit of the Zakharov system; concentration compactness methods; unique continuation},
language = {eng},
number = {1},
pages = {57-79},
publisher = {Gauthier-Villars},
title = {On the ground states of vector nonlinear Schrödinger equations},
url = {http://eudml.org/doc/76736},
volume = {65},
year = {1996},
}

TY - JOUR
AU - Colin, Thierry
AU - Weinstein, Michael I.
TI - On the ground states of vector nonlinear Schrödinger equations
JO - Annales de l'I.H.P. Physique théorique
PY - 1996
PB - Gauthier-Villars
VL - 65
IS - 1
SP - 57
EP - 79
LA - eng
KW - ground states; vector nonlinear Schrödinger equation; limit of the Zakharov system; concentration compactness methods; unique continuation
UR - http://eudml.org/doc/76736
ER -

References

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  1. [1] H. Brezis and E.H. Lieb, Minimum Action Solutions of Some Vector Field Equations, Commun. Math. Phys., Vol. 96, 1984, pp. 97-113. Zbl0579.35025MR765961
  2. [2] T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Matemáticos26, Instituto de Matemática-UFRJ Rio de Janeiro, 1993. 
  3. [3] T. Cazenave and P.-L. Lions, Orbital Stability of Standing Waves for Some Nonlinear Schrödinger equations, Comm. Math. Phys., Vol. 85, 1982, pp. 549-561. Zbl0513.35007MR677997
  4. [4] T. Colin, On the Cauchy Problem for a Nonlocal, Nonlinear Schrödinger Equation Occuring in Plasma Physics, Differential and Integral Equations, Vol. 6, n° 6, Nov. 1994, pp. 1431-1450. Zbl0780.35104MR1235204
  5. [5] T. Colin, On the Standing Waves Solutions to a Nonlocal, Nonlinear Schrödinger Equation Occuring in Plasma Physics, Physica D, 64, 1993, pp. 215-236. Zbl0780.35105MR1214553
  6. [6] L.M. Degtyarev and V.E. Zakharov, Dipole Character of the Collapse of Langmuir Waves, JETP Lett., 20, 1974, pp. 164-165. 
  7. [7] R.O. Dendy, Plasma Dynamics, Oxford University press, 1990. 
  8. [8] M. Grillakis, J. Shatah and W.A. Strauss, Stability Theory of Solitary Waves in the Presence of Symmetry I, J. Func. Anal., 74, 1987, pp. 263-272. Zbl0656.35122MR901236
  9. [9] J. Gibbons, S.G. Thornhill, M.J. Wardrop and D. Ter Harr, On the Theory of Langmuir Solitons, J. Plasma Phys., 17, 1977, pp. 153-170. 
  10. [10] B. Gidas, W.M. Ni and L. Nirenberg, Symmetry of Positive Solutions of Nonlinear Elliptic Equations in Rn, Mathematical analysis and applications, Part A, Advances in mathematics supplementary studies, 7A, 1981, pp. 369-402. Zbl0469.35052
  11. [11] J. Ginibre and G. Velo, On a Class of Nonlinear Schrödinger Equations, I the Cauchy Problem, General Case, J. Func. Anal., 32, 1979, pp. 1-32. Zbl0396.35028MR533218
  12. [12] L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. III, Springer Verlag. Zbl0601.35001MR404822
  13. [13] T. Kato, On Nonlinear Schrödinger Equations, Ann. Inst. Henri Poincaré, Physique théorique, Vol. 46, n° 1, 1987, pp. 113-129. Zbl0632.35038MR877998
  14. [14] M.K. Kwong, Uniqueness of Positive Solutions of Δu - u + up = 0 in Rn, Arch. Rat. Mech. Anal., 105, 1991, pp. 583-599. Zbl0724.34023
  15. [15] G. Le Mesurier, G.C. Papanicolaou, C. Sulem and P.-L. Sulem, The Focusing Singularity of the Nonlinear Schrödinger Equation, in Directions in Partial Differential Equations, edited by M. G. CRANDALL, P. H. RABINOWITZ and R. E. TURNER, Academic, New York, 1987, pp. 159-201. Zbl0659.35020MR1013838
  16. [16] P.-L. Lions, The Concentration Compactness Principle in the Calculus of Variations. The Locally Compact Case, Part I, Ann. Inst. Henri Poincaré, Analyse non linéaire, 1, n° 2, 1984, pp. 109-145, and Part II, Ann. Inst. Henri Poincaré, Analyse non linéaire, 1, n° 4, 1984, pp. 223-283. Zbl0541.49009
  17. [17] O. Lopes, Radial Symmetry of Minimizer for some Translation and Rotation Invariant Functionals, Preprint UNICAMP, Campinas, Brazil, to appear in Journal of Differential Equations. Zbl0842.49004MR1370147
  18. [18] T. Ozawa and Y. Tsutsumi, Existence and Smoothing Effect of Solutions for the Zakharov Equations, Publ. RIMS, Kyoto Univ., 28, 1992, pp. 329-361. Zbl0842.35116MR1184829
  19. [19] S.H. Schochet and M.I. Weinstein, The Nonlinear Schrödinger Limit of the Zakharov Equations Governing Langmuir Turbulence, Comm. Math. Phys., 106, 1986, pp. 569-580. Zbl0639.76054MR860310
  20. [20] W.A. Strauss, Existence of Solitary Waves in Higher Dimensions, Comm. Math. Phys., 55, 1977, pp. 149-162. Zbl0356.35028MR454365
  21. [21] F. Trèves, Linear Partial Differential Equations, Gordon and Breach, 1970. Zbl0209.12001
  22. [22] M.I. Weinstein, Nonlinear Schrödinger Equation and Sharp Interpolation Estimates, Comm. Math. Phys., 87, 1983, pp. 567-576. Zbl0527.35023
  23. [23] M.I. Weinstein, On the Structure and Formation of Singularities in Solutions of Nonlinear Dispersive Evolution Equations, Comm. in Partial Diff. Eqns, 11, 1986, pp. 545-565. Zbl0596.35022MR829596
  24. [24] M.I. Weinstein, Lyapunov Stability of Ground States of Nonlinear Dispersive Evolution Equations, Commun. Pure Appl. Math., 39, 1986, pp. 51-68. Zbl0594.35005MR820338
  25. [25] V.E. Zakharov, S.L. Musher and A.M. Rubenchik, Hamiltonian Approach to the Description of Nonlinear Plasma Phenomena, Physics Reports, 129, n° 5, 1985, pp. 285-366. MR824169
  26. [26] V.E. Zakharov, A.F. Mastryukov and V.S. Synakh, Two-Dimensional Collapse of Langmuir Waves, JETP Lett., 20, n° 1, July 1974. 
  27. [27] V.E. Zakharov and V.S. Synakh, The Nature of the Self-Focusing Singularity, Sov. Phys. JETP, 41, 1976, pp. 465-468. 

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