WKB Analysis for the Gross-Pitaevskii Equation With Non-Trivial Boundary Conditions at Infinity

Alazard, Thomas; Carles, RéMi

Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 3, pp. 959-977.

[1] Abdullaev F. Kh., Gammal A., Tomio L., Frederico T., Stability of Trapped Bose-Einstein Condensates, Phys. Rev. A 63 (4) (2001) 043604.

[2] Alazard T., Carles R., Semi-Classical Limit of Schrödinger-Poisson Equations in Space Dimension $n\ge 3$, J. Differential Equations 233 (1) (2007) 241-275. | MR | Zbl

[3] T. Alazard, R. Carles, Super-critical geometric optics for nonlinear Schrödinger equations, Preprint, arXiv:0704.2488. | Zbl

[4] R. Anton, Cubic nonlinear Schrödinger equation on three dimensional balls with radial data, Comm. Partial Differential Equations, in press. | MR | Zbl

[5] Anton R., Global Existence for Defocusing Cubic NLS and Gross-Pitaevskii Equations in Exterior Domains, J. Math. Pures Appl. 89 (4) (2008) 335-354. | MR | Zbl

[6] Bethuel F., Saut J.-C., Travelling Waves for the Gross-Pitaevskii Equation. I, Ann. Inst. H. Poincaré Phys. Théor. 70 (2) (1999) 147-238. | Numdam | MR | Zbl

[7] Brenier Y., Convergence of the Vlasov-Poisson System to the Incompressible Euler Equations, Comm. Partial Differential Equations 25 (3-4) (2000) 737-754. | MR | Zbl

[8] Carles R., Geometric Optics and Instability for Semi-Classical Schrödinger Equations, Arch. Ration. Mech. Anal. 183 (3) (2007) 525-553. | MR | Zbl

[9] Carles R., WKB Analysis for Nonlinear Schrödinger Equations With Potential, Commun. Math. Phys. 269 (1) (2007) 195-221. | MR | Zbl

[10] Colin T., Soyeur A., Some Singular Limits for Evolutionary Ginzburg-Landau Equations, Asymptotic Anal. 13 (4) (1996) 361-372. | MR | Zbl

[11] Dalfovo F., Giorgini S., Pitaevskii L. P., Stringari S., Theory of Bose-Einstein Condensation in Trapped Gases, Rev. Mod. Phys. 71 (3) (1999) 463-512.

[12] Gallo C., Schrödinger Group on Zhidkov Spaces, Adv. Differential Equations 9 (5-6) (2004) 509-538. | MR | Zbl

[13] C. Gallo, The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity, Preprint. | MR | Zbl

[14] Gammal A., Frederico T., Tomio L., Chomaz Ph., Atomic Bose-Einstein Condensation With Three-Body Interactions and Collective Excitations, J. Phys. B 33 (2000) 4053-4067.

[15] Gasser I., Lin C.-K., Markowich P. A., A Review of Dispersive Limits of (Non)Linear Schrödinger-Type Equations, Taiwanese J. Math. 4 (4) (2000) 501-529. | MR | Zbl

[16] P. Gérard, Remarques sur l'analyse semi-classique de l'équation de Schrödinger non linéaire, in: Séminaire sur les Équations aux Dérivées Partielles, 1992-1993, École polytech., Palaiseau, 1993, Exp. No. XIII. | Numdam | Zbl

[17] Gérard P., The Cauchy Problem for the Gross-Pitaevskii Equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (5) (2006) 765-779. | Numdam | Zbl

[18] Grenier E., Semiclassical Limit of the Nonlinear Schrödinger Equation in Small Time, Proc. Amer. Math. Soc. 126 (2) (1998) 523-530. | Zbl

[19] Josserand C., Pomeau Y., Nonlinear Aspects of the Theory of Bose-Einstein Condensates, Nonlinearity 14 (5) (2001) R25-R62. | Zbl

[20] Lannes D., Sharp Estimates for Pseudo-Differential Operators With Symbols of Limited Smoothness and Commutators, J. Funct. Anal. 232 (2) (2006) 495-539. | Zbl

[21] Lin F., Zhang P., Semiclassical Limit of the Gross-Pitaevskii Equation in an Exterior Domain, Arch. Rational Mech. Anal. 179 (1) (2006) 79-107. | MR | Zbl

[22] Majda A., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. | MR | Zbl

[23] Métivier G., Remarks on the Well-Posedness of the Nonlinear Cauchy Problem, in: Geometric Analysis of PDE and Several Complex Variables, Contemp. Math., vol. 368, Amer. Math. Soc., Providence, RI, 2005, pp. 337-356. | MR | Zbl

[24] Michinel H., Campo-Táboas J., García-Fernández R., Salgueiro J. R., Quiroga-Teixeiro M. L., Liquid Light Condensates, Phys. Rev. E 65 (2002) 066604.

[25] Pitaevskii L., Stringari S., Bose-Einstein Condensation, International Series of Monographs on Physics, vol. 116, The Clarendon Press, Oxford University Press, Oxford, 2003. | MR | Zbl

[26] Sideris T., Formation of Singularities in Three-Dimensional Compressible Fluids, Commun. Math. Phys. 101 (1985) 475-485. | MR | Zbl

[27] Taylor M., Partial Differential Equations. III, Nonlinear Equations, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997. | MR | Zbl

[28] Thomann L., Instabilities for Supercritical Schrödinger Equations in Analytic Manifolds, J. Differential Equations 245 (1) (2008) 249-280. | MR | Zbl

[29] P.E. Zhidkov, The Cauchy problem for a nonlinear Schrödinger equation, JINR Commun., P5-87-373, Dubna, 1987 (in Russian).

[30] Zhidkov P. E., Korteweg-De Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Mathematics, vol. 1756, Springer-Verlag, Berlin, 2001. | MR | Zbl