On the ground states of vector nonlinear Schrödinger equations
Annales de l'I.H.P. Physique théorique, Volume 65 (1996) no. 1, pp. 57-79.
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     author = {Colin, Thierry and Weinstein, Michael I.},
     title = {On the ground states of vector nonlinear {Schr\"odinger} equations},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     pages = {57--79},
     publisher = {Gauthier-Villars},
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     number = {1},
     year = {1996},
     zbl = {0863.35101},
     mrnumber = {1407166},
     language = {en},
     url = {http://archive.numdam.org/item/AIHPA_1996__65_1_57_0/}
}
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Colin, Thierry; Weinstein, Michael I. On the ground states of vector nonlinear Schrödinger equations. Annales de l'I.H.P. Physique théorique, Volume 65 (1996) no. 1, pp. 57-79. http://archive.numdam.org/item/AIHPA_1996__65_1_57_0/

[1] H. Brezis and E.H. Lieb, Minimum Action Solutions of Some Vector Field Equations, Commun. Math. Phys., Vol. 96, 1984, pp. 97-113. | MR | Zbl

[2] T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Matemáticos 26, Instituto de Matemática-UFRJ Rio de Janeiro, 1993.

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[6] L.M. Degtyarev and V.E. Zakharov, Dipole Character of the Collapse of Langmuir Waves, JETP Lett., 20, 1974, pp. 164-165.

[7] R.O. Dendy, Plasma Dynamics, Oxford University press, 1990.

[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. | MR | Zbl

[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] 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. | Zbl

[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. | MR | Zbl

[12] L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. III, Springer Verlag. | MR | Zbl

[13] T. Kato, On Nonlinear Schrödinger Equations, Ann. Inst. Henri Poincaré, Physique théorique, Vol. 46, n° 1, 1987, pp. 113-129. | Numdam | MR | Zbl

[14] M.K. Kwong, Uniqueness of Positive Solutions of Δu - u + up = 0 in Rn, Arch. Rat. Mech. Anal., 105, 1991, pp. 583-599.

[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. | MR | Zbl

[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. | Numdam | Zbl

[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. | MR

[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. | MR | Zbl

[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. | MR | Zbl

[20] W.A. Strauss, Existence of Solitary Waves in Higher Dimensions, Comm. Math. Phys., 55, 1977, pp. 149-162. | MR | Zbl

[21] F. Trèves, Linear Partial Differential Equations, Gordon and Breach, 1970. | Zbl

[22] M.I. Weinstein, Nonlinear Schrödinger Equation and Sharp Interpolation Estimates, Comm. Math. Phys., 87, 1983, pp. 567-576. | Zbl

[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. | MR | Zbl

[24] M.I. Weinstein, Lyapunov Stability of Ground States of Nonlinear Dispersive Evolution Equations, Commun. Pure Appl. Math., 39, 1986, pp. 51-68. | MR | Zbl

[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. | MR

[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] V.E. Zakharov and V.S. Synakh, The Nature of the Self-Focusing Singularity, Sov. Phys. JETP, 41, 1976, pp. 465-468.