On the ground states of vector nonlinear Schrödinger equations
Colin, Thierry ; Weinstein, Michael I.
Annales de l'I.H.P. Physique théorique, Tome 65 (1996), p. 57-79 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)
Publié le : 1996-01-01
@article{AIHPA_1996__65_1_57_0,
     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},
     volume = {65},
     year = {1996},
     pages = {57-79},
     mrnumber = {1407166},
     zbl = {0863.35101},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPA_1996__65_1_57_0}
}

Colin, Thierry; Weinstein, Michael I. On the ground states of vector nonlinear Schrödinger equations. Annales de l'I.H.P. Physique théorique, Tome 65 (1996) pp. 57-79. http://gdmltest.u-ga.fr/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 765961 | Zbl 0579.35025

[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 677997 | Zbl 0513.35007

[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 1235204 | Zbl 0780.35104

[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 1214553 | Zbl 0780.35105

[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 901236 | Zbl 0656.35122

[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 0469.35052

[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 533218 | Zbl 0396.35028

[12] L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. III, Springer Verlag. | MR 404822 | Zbl 0601.35001

[13] T. Kato, On Nonlinear Schrödinger Equations, Ann. Inst. Henri Poincaré, Physique théorique, Vol. 46, n° 1, 1987, pp. 113-129. | Numdam | MR 877998 | Zbl 0632.35038

[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 1013838 | Zbl 0659.35020

[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 0541.49009

[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 1370147

[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 1184829 | Zbl 0842.35116

[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 860310 | Zbl 0639.76054

[20] W.A. Strauss, Existence of Solitary Waves in Higher Dimensions, Comm. Math. Phys., 55, 1977, pp. 149-162. | MR 454365 | Zbl 0356.35028

[21] F. Trèves, Linear Partial Differential Equations, Gordon and Breach, 1970. | Zbl 0209.12001

[22] M.I. Weinstein, Nonlinear Schrödinger Equation and Sharp Interpolation Estimates, Comm. Math. Phys., 87, 1983, pp. 567-576. | Zbl 0527.35023

[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 829596 | Zbl 0596.35022

[24] M.I. Weinstein, Lyapunov Stability of Ground States of Nonlinear Dispersive Evolution Equations, Commun. Pure Appl. Math., 39, 1986, pp. 51-68. | MR 820338 | Zbl 0594.35005

[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 824169

[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.