Stability of Solitary Waves for a System of Nonlinear Schrödinger Equations With Three Wave Interaction
Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 6, pp. 2211-2226.
@article{AIHPC_2009__26_6_2211_0,
     author = {Colin, M. and Colin, Th. and Ohta, M.},
     title = {Stability of {Solitary} {Waves} for a {System} of {Nonlinear} {Schr\"odinger} {Equations} {With} {Three} {Wave} {Interaction}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {2211--2226},
     publisher = {Elsevier},
     volume = {26},
     number = {6},
     year = {2009},
     doi = {10.1016/j.anihpc.2009.01.011},
     mrnumber = {2569892},
     zbl = {1180.35478},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2009.01.011/}
}
TY  - JOUR
AU  - Colin, M.
AU  - Colin, Th.
AU  - Ohta, M.
TI  - Stability of Solitary Waves for a System of Nonlinear Schrödinger Equations With Three Wave Interaction
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2009
SP  - 2211
EP  - 2226
VL  - 26
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2009.01.011/
DO  - 10.1016/j.anihpc.2009.01.011
LA  - en
ID  - AIHPC_2009__26_6_2211_0
ER  - 
%0 Journal Article
%A Colin, M.
%A Colin, Th.
%A Ohta, M.
%T Stability of Solitary Waves for a System of Nonlinear Schrödinger Equations With Three Wave Interaction
%J Annales de l'I.H.P. Analyse non linéaire
%D 2009
%P 2211-2226
%V 26
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2009.01.011/
%R 10.1016/j.anihpc.2009.01.011
%G en
%F AIHPC_2009__26_6_2211_0
Colin, M.; Colin, Th.; Ohta, M. Stability of Solitary Waves for a System of Nonlinear Schrödinger Equations With Three Wave Interaction. Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 6, pp. 2211-2226. doi : 10.1016/j.anihpc.2009.01.011. http://www.numdam.org/articles/10.1016/j.anihpc.2009.01.011/

[1] Berestycki H., Cazenave T., Instabilité Des États Stationnaires Dans Les Équations De Schrödinger Et De Klein-Gordon Non Linéaires, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981) 489-492. | MR | Zbl

[2] Cazenave T., Semilinear Schrödinger Equations, Courant Lect. Notes Math. 10, Amer. Math. Soc., 2003. | MR | Zbl

[3] Cazenave T., Lions P. L., Orbital Stability of Standing Waves for Some Nonlinear Schrödinger Equations, Comm. Math. Phys. 85 (1982) 549-561. | MR | Zbl

[4] Chang S.-M., Gustafson S., Nakanishi K., Tsai T.-P., Spectra of Linearized Operators for NLS Solitary Waves, SIAM J. Math. Anal. 39 (2007/2008) 1070-1111. | MR | Zbl

[5] Colin M., Colin T., On a Quasi-Linear Zakharov System Describing Laser-Plasma Interactions, Differential Integral Equations 17 (2004) 297-330. | MR | Zbl

[6] Colin M., Colin T., A Numerical Model for the Raman Amplification for Laser-Plasma Interaction, J. Comput. Appl. Math. 193 (2006) 535-562. | MR | Zbl

[7] De Bouard A., Instability of Stationary Bubbles, SIAM J. Math. Anal. 26 (1995) 566-582. | MR | Zbl

[8] Di Menza L., Gallo C., The Black Solitons of One-Dimensional NLS Equations, Nonlinearity 20 (2007) 461-496. | MR | Zbl

[9] Esteban M., Strauss W., Nonlinear Bound States Outside an Insulated Sphere, Comm. Partial Differential Equations 19 (1994) 177-197. | MR | Zbl

[10] Fukuizumi R., Remarks on the Stable Standing Waves for Nonlinear Schrödinger Equations With Double Power Nonlinearity, Adv. Math. Sci. Appl. 13 (2003) 549-564. | MR | Zbl

[11] Gesztesy F., Jones C. K.R. T., Latushkin Y., Stanislavova M., A Spectral Mapping Theorem and Invariant Manifolds for Nonlinear Schrödinger Equations, Indiana Univ. Math. J. 49 (2000) 221-243. | MR | Zbl

[12] 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

[13] 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

[14] Iliev I. D., Kirchev P., Stability and Instability of Solitary Waves for One-Dimensional Singular Schrödinger Equations, Differential Integral Equations 6 (1993) 685-703. | MR | Zbl

[15] Kwong M. K., Uniqueness of Positive Solutions of Δu-u+u p =0 in R n , Arch. Ration. Mech. Anal. 105 (1989) 234-266. | MR | Zbl

[16] Lieb E. H., Loss M., Analysis, Grad. Stud. Math., vol. 14, second ed., Amer. Math. Soc., 2001. | MR | Zbl

[17] Mizumachi T., A Remark on Linearly Unstable Standing Wave Solutions to NLS, Nonlinear Anal. 64 (2006) 657-676. | MR | Zbl

[18] Shatah J., Strauss W., Instability of Nonlinear Bound States, Comm. Math. Phys. 100 (1985) 173-190. | MR | Zbl

[19] Shatah J., Strauss W., Spectral Condition for Instability, in: Contemp. Math., vol. 255, 2000, pp. 189-198. | MR | Zbl

[20] Sulem C., Sulem P.-L., The Nonlinear Schrödinger Equation: Self-Focusing and Wave-Collapse, Appl. Math. Sci., vol. 139, Springer-Verlag, 1999. | MR | Zbl

[21] Weinstein M. I., Lyapunov Stability of Ground States of Nonlinear Dispersive Evolution Equations, Comm. Pure Appl. Math. 39 (1986) 51-68. | MR | Zbl

[22] Weinstein M. I., Modulational Stability of Ground States of Nonlinear Schrödinger Equations, SIAM J. Math. Anal. 16 (1985) 472-491. | MR | Zbl

Cité par Sources :