Stability of standing waves for nonlinear Schrödinger equations with potentials
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2003-2004), Exposé no. 9, 8 p.
@article{SEDP_2003-2004____A9_0,
author = {Fukuizumi, Reika},
title = {Stability of standing waves for nonlinear  {Schr\"odinger} equations with potentials},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
note = {talk:9},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2003-2004},
mrnumber = {2117041},
language = {en},
url = {http://www.numdam.org/item/SEDP_2003-2004____A9_0/}
}
TY  - JOUR
AU  - Fukuizumi, Reika
TI  - Stability of standing waves for nonlinear  Schrödinger equations with potentials
JO  - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
N1  - talk:9
PY  - 2003-2004
DA  - 2003-2004///
PB  - Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://www.numdam.org/item/SEDP_2003-2004____A9_0/
UR  - https://www.ams.org/mathscinet-getitem?mr=2117041
LA  - en
ID  - SEDP_2003-2004____A9_0
ER  - 
Fukuizumi, Reika. Stability of standing waves for nonlinear  Schrödinger equations with potentials. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2003-2004), Exposé no. 9, 8 p. http://www.numdam.org/item/SEDP_2003-2004____A9_0/

[1] G. Baym and C. J. Pethick, “Ground-state properties of magnetically trapped Bose-condensed rubidium gas”, Phys. Rev. Lett. Vol. 76 (1996), 6–9.

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

[3] T. Cazenave, “An introduction to nonlinear Schrödinger equations,” Textos de Métods Matemáticos 26, IM-UFRJ, Rio de Janeiro, 1993.

[4] T. Cazenave and P. L. Lions, “Orbital stability of standing waves for some nonlinear Schrödinger equations”, Commun. Math. Phys. Vol. 85 (1982), 549–561. | Zbl 0513.35007

[5] R. Y. Chiao, E. Garmine and C. H. Townes, “Self-trapping of optical beams”, Phys. Rev. Lett. Vol. 13 (1964), 479–482.

[6] C. Cid and P. Felmer, “Orbital stability of standing waves for the nonlinear Schrödinger equation with potential”, Rev. Math. Phys. Vol. 13 (2001), 1529–1546. | Zbl 1038.35112

[7] G. Fibich and X. P. Wang, “Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities”, Physica D. Vol. 175 (2003), 96–108. | Zbl 1098.74614

[8] R. Fukuizumi, “Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential”, Discrete Contin. Dynam. Systems. Vol. 7 (2001), 525–544. | Zbl 0992.35094

[9] R. Fukuizumi and M. Ohta, “Instability of standing waves for nonlinear Schrödinger equations with potentials”, Differential and Integral Eqs. Vol. 16 (2003), 691–706. | Zbl 1031.35132

[10] R. Fukuizumi and M. Ohta, “Stability of standing waves for nonlinear Schrödinger equations with potentials”, Differential and Integral Eqs. Vol. 16 (2003), 111–128. | Zbl 1031.35131

[11] R. Fukuizumi, “Stability of standing waves for nonlinear Schrödinger equations with critical power nonlinearity and potentials,” Preprint. | Zbl 1107.35100

[12] A. Griffin, D. W. Snoke and S. Stringari, “Bose-Einstein condensation,” Cambridge University Press, Cambridge, 1995.

[13] M. Grillakis, J. Shatah and W. Strauss, “Stability theory of solitary waves in the presence of symmetry I”, J. Funct. Anal. Vol. 74 (1987), 160–197. | Zbl 0656.35122

[14] M. Grillakis, J. Shatah and W. Strauss, “Stability theory of solitary waves in the presence of symmetry II”, J. Funct. Anal. Vol. 94 (1990), 308–348. | Zbl 0711.58013

[15] M. Hirose and M. Ohta, “Structure of positive radial solutions to scalar field equations with harmonic potential”, J. Differential Eqs. Vol. 178 (2002), 519–540. | Zbl 1011.34039

[16] M. Hirose and M. Ohta, “Uniqueness of positive solutions to scalar field equations with harmonic potential”, Preprint.

[17] Y. Kabeya and K. Tanaka, “Uniqueness of positive radial solutions of semilinear elliptic equations in ${ℝ}^{n}$ and Séré’s non-degeneracy condition”, Commun. Partial. Differential. Eqs. Vol. 24 (1999), 563–598. | Zbl 0930.35064

[18] M. Kunze, T. Küpper, V. K. Mezentsev, E. G. Shapiro and S. Turitsyn, “Nonlinear solitary waves with Gaussian tails”, Physica D. Vol. 128 (1999), 273–295. | Zbl 0935.35152

[19] M. K. Kwong, “Uniqueness of positive solutions of $\Delta u-u+{u}^{p}=0$ in ${ℝ}^{n}$”, Arch. Rational Mech. Anal. Vol. 105 (1989), 234–266. | Zbl 0676.35032

[20] Y. Li and W. N. Ni, “Radial symmetry of positive solutions nonlinear elliptic equations in ${ℝ}^{n}$”, Commun. Partial Differential. Eqs. Vol. 18 (1993), 1043–1054. | Zbl 0788.35042

[21] Y. G. Oh, “Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials”, Commun. Math. Phys. Vol. 121 (1989), 11–33. | Zbl 0693.35132

[22] Y. G. Oh, “Cauchy problem and Ehrenfest’s law of nonlinear Schrödinger equtions with potentials”, J. Differential Eqs. Vol. 81 (1989), 255–274. | Zbl 0703.35158

[23] H. A. Rose and M. I. Weinstein, “On the bound states of the nonlinear Schrödinger equation with a linear potential”, Phyica D. Vol. 30 (1988), 207–218. | Zbl 0694.35202

[24] J. Shatah, “Stable standing waves of nonlinear Klein-Gordon equations”, Commun. Math. Phys. Vol. 91 (1983), 313–327. | Zbl 0539.35067

[25] J. Shatah and W. Strauss, “Instability of nonlinear bound states”, Commun. Math. Phys. Vol. 100 (1985), 173–190. | Zbl 0603.35007

[26] C. Sulem and P.-L. Sulem, “The nonlinear Schrödinger equation. Self-focusing and wave collapse,” Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999. | Zbl 0928.35157

[27] M. Wadati and T. Tsurumi, “Collapses of wavefunctions in multi-dimensional nonlinear Schrödinger equations under harmonic potential”, J. Phys. Soc. Japan. Vol. 66 (1997), 3031–3034. | Zbl 0973.76623

[28] M. I. Weinstein, “Nonlinear Schrödinger equations and sharp interpolation estimates”, Comm. Math. Phys. Vol. 87 (1983), 567–576. | Zbl 0527.35023

[29] V. E. Zakharov, “Collapse of langmuir waves”, Sov. Phys. JETP Vol. 35 (1972), 908–912.

[30] J. Zhang, “Stability of standing waves for the nonlinear Schrödinger equations with unbounded potentials”, Z. Angew. Math. Phys. Vol. 51 (2000), 489–503. | Zbl 0985.35085

[31] J. Zhang, “Stability of Attractive Bose-Einstein Condensates”, Journal of Statistical Physics. Vol. 101 (2000), 731–745. | Zbl 0989.82024

[32] J. Zhang, “Sharp criteria for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential”, (1999), Preprint.