Stability of solitary waves for derivative nonlinear Schrödinger equation

Colin, Mathieu; Ohta, Masahito

doi:10.1016/j.anihpc.2005.09.003

Colin, Mathieu ; Ohta, Masahito

Annales de l'I.H.P. Analyse non linéaire, Tome 23 (2006) no. 5, pp. 753-764.

MR Zbl

DOI : 10.1016/j.anihpc.2005.09.003

@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},
     pages = {753--764},
     publisher = {Elsevier},
     volume = {23},
     number = {5},
     year = {2006},
     doi = {10.1016/j.anihpc.2005.09.003},
     mrnumber = {2259615},
     zbl = {1104.35050},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2005.09.003/}
}

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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) no. 5, pp. 753-764. doi : 10.1016/j.anihpc.2005.09.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2005.09.003/

Bibliographie
Cité par

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

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

[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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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

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