Stability of standing waves for nonlinear Schrödinger equations with potentials

Fukuizumi, Reika

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2003-2004), Exposé no. 9, 8 p.

[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

[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

[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

[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

[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

[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

[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

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

[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

[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

[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

[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

[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

[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

[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

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

[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

[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

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

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

[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

[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

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

[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

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

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