Geodesic
Orbital stability of semitrivial standing waves for the Klein–Gordon–Schrödinger system
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 2, pp. 315-323.

Voir la notice de l'article provenant de la source Numdam

corrigé par Erratum

In the present paper, we study the orbital stability and instability of standing waves of the Klein–Gordon–Schrödinger system. Especially, we are interested in a standing wave which is expressed by the unique positive solution w 1 to a certain scalar field equation. By utilizing the property of the positive solution w 1 , we can apply the general theory of Grillakis, Shatah and Strauss (1987) [11] and show the stability and instability of the standing wave.

@article{AIHPC_2011__28_2_315_0,
     author = {Kikuchi, Hiroaki},
     title = {Orbital stability of semitrivial standing waves for the {Klein{\textendash}Gordon{\textendash}Schr\"odinger} system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {315--323},
     publisher = {Elsevier},
     volume = {28},
     number = {2},
     year = {2011},
     doi = {10.1016/j.anihpc.2011.02.003},
     mrnumber = {2784074},
     zbl = {1216.35116},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2011.02.003/}
}
TY  - JOUR
AU  - Kikuchi, Hiroaki
TI  - Orbital stability of semitrivial standing waves for the Klein–Gordon–Schrödinger system
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2011
SP  - 315
EP  - 323
VL  - 28
IS  - 2
PB  - Elsevier
UR  - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2011.02.003/
DO  - 10.1016/j.anihpc.2011.02.003
LA  - en
ID  - AIHPC_2011__28_2_315_0
ER  - 
%0 Journal Article
%A Kikuchi, Hiroaki
%T Orbital stability of semitrivial standing waves for the Klein–Gordon–Schrödinger system
%J Annales de l'I.H.P. Analyse non linéaire
%D 2011
%P 315-323
%V 28
%N 2
%I Elsevier
%U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2011.02.003/
%R 10.1016/j.anihpc.2011.02.003
%G en
%F AIHPC_2011__28_2_315_0
Kikuchi, Hiroaki. Orbital stability of semitrivial standing waves for the Klein–Gordon–Schrödinger system. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 2, pp. 315-323. doi : 10.1016/j.anihpc.2011.02.003. http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2011.02.003/

[1] J. Angulo, F. Linares, Periodic pulses of coupled nonlinear Schrödinger equations in optics, Indiana Univ. Math. J. 56 (2007), 847-877 | MR | Zbl

[2] A. Bachelot, Problème de Cauchy pour des systèmes hyperboliques semi-linéaires, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 453-478 | MR | EuDML | Zbl | mathdoc-id

[3] J.B. Baillon, J.M. Chadam, The Cauchy problem for the coupled Klein–Gordon–Schrödinger equations, Contemporary Developments in Continuum Mechanics and Partial Differential Equations, North-Holland, Amsterdam (1978), 37-44 | MR | Zbl

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

[5] H. Berestycki, P.L. Lions, Nonlinear scalar field equation I, Arch. Rat. Mech. Anal. 82 (1983), 313-346 | MR

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

[7] I. Fukuda, M. Tsutsumi, On coupled Klein–Gordon–Schrödinger equations II, J. Math. Anal. Appl. 66 (1978), 358-378 | MR | Zbl

[8] B. Gidas, W.M. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in R n , Adv. Math. Suppl. Stud. vol. 7A, Academic Press (1981), 369-402 | MR

[9] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer (1983) | MR | Zbl

[10] M. Grillakis, Linearized instability for nonlinear Schrödinger and Klein–Gordon equations, Comm. Pure Appl. Math. 41 (1988), 747-774 | MR | Zbl

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

[12] A. Grünrock, H. Pecher, Bounds in time for the Klein–Gordon–Schrödinger and the Zakharov systems, Hokkaido Math. J. 35 (2006), 139-153 | MR | Zbl

[13] N. Hayashi, W. Von Wahl, On the global strong solutions of coupled Klein–Gordon–Schrödinger equations, J. Math. Soc. Japan 39 (1987), 489-497 | MR | Zbl

[14] H. Kikuchi, M. Ohta, Instability of standing waves for the Klein–Gordon–Schrödinger system, Hokkaido Math. J. 37 (2008), 735-748 | MR | Zbl

[15] H. Kikuchi, M. Ohta, Stability of standing waves for the Klein–Gordon–Schrödinger system, J. Math. Anal. Appl. 365 (2010), 109-114 | MR | Zbl

[16] M. Kwong, Uniqueness of positive solutions of Δu-u+u p =0 in R n , Arch. Rat. Mech. Anal. 105 (1989), 243-266 | MR | Zbl

[17] W.-M. Ni, I. Takagi, Locating the peaks of least-energy solution to a semilinear Neumann problem, Duke Math. J. 70 (1993), 247-281 | MR | Zbl

[18] H. Pecher, Global solutions of the Klein–Gordon–Schrödinger system with rough data, Differential Integral Equations 17 (2004), 179-214 | MR | Zbl

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

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

[21] M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/1983), 567-576 | MR | Zbl

[22] M. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Appl. Math. 16 (1985), 472-490 | MR | Zbl

[23] A.I. Yew, Stability analysis of multipulses in nonlinearly-coupled Schödinger equations, Indiana Univ. Math. J. 49 (2000), 1079-1124 | MR | Zbl

Cité par Sources :