Stability of solitary waves for derivative nonlinear Schrödinger equation
Colin, Mathieu ; Ohta, Masahito
Annales de l'I.H.P. Analyse non linéaire, Tome 23 (2006), p. 753-764 / Harvested from Numdam
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@article{AIHPC_2006__23_5_753_0,
     author = {Colin, Mathieu and Ohta, Masahito},
     title = {Stability of solitary waves for derivative nonlinear Schr\"odinger equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {23},
     year = {2006},
     pages = {753-764},
     doi = {10.1016/j.anihpc.2005.09.003},
     mrnumber = {2259615},
     zbl = {1104.35050},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2006__23_5_753_0}
}

Colin, Mathieu; Ohta, Masahito. Stability of solitary waves for derivative nonlinear Schrödinger equation. Annales de l'I.H.P. Analyse non linéaire, Tome 23 (2006) pp. 753-764. doi : 10.1016/j.anihpc.2005.09.003. http://gdmltest.u-ga.fr/item/AIHPC_2006__23_5_753_0/

[1] Biagioni H.A., Linares F., Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc. 353 (2001) 3649-3659. | MR 1837253 | Zbl 0970.35154

[2] Berestycki H., Cazenave T., Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris 293 (1981) 489-492. | MR 646873 | Zbl 0492.35010

[3] Brézis H., Lieb E.H., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983) 486-490. | MR 699419 | Zbl 0526.46037

[4] Cazenave T., Semilinear Schrödinger equations, Courant Lecture Notes Math., vol. 10, New York University, Courant Institute of Mathematical Sciences, American Mathematical Society, New York, Providence, RI, 2003. | MR 2002047 | Zbl 1055.35003

[5] Cazenave T., Lions P.L., Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982) 549-561. | MR 677997 | Zbl 0513.35007

[6] Fröhlich J., Lieb E.H., Loss M., Stability of Coulomb systems with magnetic fields I. The one-electron atom, Comm. Math. Phys. 104 (1986) 251-270. | MR 836003 | Zbl 0595.35098

[7] Grillakis M., Shatah J., Strauss W., Stability theory of solitary waves in the presence of symmetry, I, J. Funct. Anal. 74 (1987) 160-197. | MR 901236 | Zbl 0656.35122

[8] Grillakis M., Shatah J., Strauss W., Stability theory of solitary waves in the presence of symmetry, II, J. Funct. Anal. 94 (1990) 308-348. | MR 1081647 | Zbl 0711.58013

[9] Guo Boling, Wu Yaping, Orbital stability of solitary waves for the nonlinear derivative Schrödinger equation, J. Differential Equations 123 (1995) 35-55. | MR 1359911 | Zbl 0844.35116

[10] Hayashi N., The initial value problem for the derivative nonlinear Schrödinger equation in the energy space, Nonlinear Anal. 20 (1993) 823-833. | MR 1214746 | Zbl 0787.35099

[11] Hayashi N., Ozawa T., On the derivative nonlinear Schrödinger equation, Physica D 55 (1992) 14-36. | MR 1152001 | Zbl 0741.35081

[12] Hayashi N., Ozawa T., Finite energy solutions of nonlinear Schrödinger equations of derivative type, SIAM J. Math. Anal. 25 (1994) 1488-1503. | MR 1302158 | Zbl 0809.35124

[13] Lieb E.H., On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math. 74 (1983) 441-448. | MR 724014 | Zbl 0538.35058

[14] Mio W., Ogino T., Minami K., Takeda S., Modified nonlinear Schrödinger equation for Alfvén waves propagating along the magnetic field in cold plasmas, J. Phys. Soc. Japan 41 (1976) 265-271. | MR 462141

[15] Mjølhus E., On the modulational instability of hydromagnetic waves parallel to the magnetic field, J. Plasma Phys. 16 (1976) 321-334.

[16] Nawa H., Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity, J. Math. Soc. Japan 46 (1994) 557-586. | MR 1291107 | Zbl 0829.35121

[17] Ohta M., Stability of standing waves for the generalized Davey-Stewartson system, J. Dynam. Differential Equations 6 (1994) 325-334. | MR 1280142 | Zbl 0805.35098

[18] Ohta M., Stability and instability of standing waves for one-dimensional nonlinear Schrödinger equations with double power nonlinearity, Kodai Math. J. 18 (1995) 68-74. | MR 1317007 | Zbl 0868.35111

[19] Ohta M., Stability and instability of standing waves for the generalized Davey-Stewartson system, Differential Integral Equations 8 (1995) 1775-1788. | MR 1347979 | Zbl 0827.35122

[20] Ohta M., Blow-up solutions and strong instability of standing waves for the generalized Davey-Stewartson system in R 2 , Ann. Inst. H. Poincaré Phys. Théor. 63 (1995) 111-117. | Numdam | MR 1354441 | Zbl 0832.35132

[21] Ozawa T., On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J. 45 (1996) 137-163. | MR 1406687 | Zbl 0859.35117

[22] Shatah J., Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys. 91 (1983) 313-327. | MR 723756 | Zbl 0539.35067

[23] Shatah J., Strauss W., Instability of nonlinear bound states, Comm. Math. Phys. 100 (1985) 173-190. | MR 804458 | Zbl 0603.35007

[24] Takaoka H., Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Differential Equations 4 (1999) 561-580. | MR 1693278 | Zbl 0951.35125

[25] Takaoka H., Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces, Electron. J. Differential Equations 2001 (42) (2001) 1-23. | MR 1836810 | Zbl 0972.35140

[26] Weinstein M.I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983) 567-576. | MR 691044 | Zbl 0527.35023

[27] Weinstein M.I., Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 (1986) 51-68. | MR 820338 | Zbl 0594.35005