Free Access
Volume 45, Number 5, September-October 2011
Page(s) 981 - 1008
Published online 10 June 2011
  1. F.Kh. Abdullaev, A. Gammal, L. Tomio and T. Frederico, Stability of trapped Bose-Einstein condensates. Phys. Rev. A 63 (2001) 043604. [CrossRef] [Google Scholar]
  2. T. Alazard and R. Carles, Semi-classical limit of Schrödinger-Poisson equations in space dimension Formula . J. Diff. Eq. 233 (2007) 241–275. [CrossRef] [Google Scholar]
  3. T. Alazard and R. Carles, Supercritical geometric optics for nonlinear Schrödinger equations. Arch. Rational Mech. Anal. 194 (2009) 315–347. [CrossRef] [Google Scholar]
  4. T. Alazard and R. Carles, WKB analysis for the Gross-Pitaevskii equation with non-trivial boundary conditions at infinity. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 959–977. [Google Scholar]
  5. W. Bao, S. Jin and P.A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime. J. Comput. Phys. 175 (2002) 487–524. [CrossRef] [MathSciNet] [Google Scholar]
  6. W. Bao, S. Jin and P.A. Markowich, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes. SIAM J. Sci. Comput. 25 (2003) 27–64. [CrossRef] [MathSciNet] [Google Scholar]
  7. C. Besse, A relaxation scheme for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 42 (2004) 934–952. [CrossRef] [MathSciNet] [Google Scholar]
  8. C. Besse, B. Bidégaray and S. Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40 (2002) 26–40. [CrossRef] [MathSciNet] [Google Scholar]
  9. Y. Brenier and L. Corrias, A kinetic formulation for multi-branch entropy solutions of scalar conservation laws. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 15 (1998) 169–190. [Google Scholar]
  10. R. Carles, Geometric optics and instability for semi-classical Schrödinger equations. Arch. Rational Mech. Anal. 183 (2007) 525–553. [CrossRef] [MathSciNet] [Google Scholar]
  11. R. Carles, Semi-classical analysis for nonlinear Schrödinger equations. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2008). [Google Scholar]
  12. R. Carles and L. Gosse, Numerical aspects of nonlinear Schrödinger equations in the presence of caustics. Math. Models Methods Appl. Sci. 17 (2007) 1531–1553. [CrossRef] [MathSciNet] [Google Scholar]
  13. T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics 10. New York University Courant Institute of Mathematical Sciences, New York (2003). [Google Scholar]
  14. J.-Y. Chemin, Dynamique des gaz à masse totale finie. Asymptotic Anal. 3 (1990) 215–220. [MathSciNet] [Google Scholar]
  15. D. Chiron and F. Rousset, Geometric optics and boundary layers for nonlinear Schrödinger equations. Comm. Math. Phys. 288 (2009) 503–546. [CrossRef] [MathSciNet] [Google Scholar]
  16. F. Dalfovo, S. Giorgini, L.P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys. 71 (1999) 463–512. [Google Scholar]
  17. P. Degond, S. Gallego and F. Méhats, An asymptotic preserving scheme for the Schrödinger equation in the semiclassical limit. C.R. Math. Acad. Sci. Paris 345 (2007) 531–536. [MathSciNet] [Google Scholar]
  18. P. Degond, S. Jin and M. Tang, On the time splitting spectral method for the complex Ginzburg-Landau equation in the large time and space scale limit. SIAM J. Sci. Comput. 30 (2008) 2466–2487. [CrossRef] [MathSciNet] [Google Scholar]
  19. J.J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27 (1974) 207–281. [CrossRef] [MathSciNet] [Google Scholar]
  20. A. Gammal, T. Frederico, L. Tomio and Ph. Chomaz, Atomic Bose-Einstein condensation with three-body intercations and collective excitations. J. Phys. B 33 (2000) 4053–4067. [CrossRef] [Google Scholar]
  21. C.L. Gardner, The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54 (1994) 409–427. [CrossRef] [MathSciNet] [Google Scholar]
  22. P. Gérard, Remarques sur l'analyse semi-classique de l'équation de Schrödinger non linéaire, Séminaire sur les Équations aux Dérivées Partielles, 1992–1993. École Polytech., Palaiseau (1993),, pp. Exp. No. XIII, 13. [Google Scholar]
  23. P. Gérard, P.A. Markowich, N.J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50 (1997) 323–379. [CrossRef] [MathSciNet] [Google Scholar]
  24. J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I The Cauchy problem, general case. J. Funct. Anal. 32 (1979) 1–32. [CrossRef] [MathSciNet] [Google Scholar]
  25. L. Gosse, Using Formula -branch entropy solutions for multivalued geometric optics computations. J. Comput. Phys. 180 (2002) 155–182. [CrossRef] [MathSciNet] [Google Scholar]
  26. L. Gosse, A case study on the reliability of multiphase WKB approximation for the one-dimensional Schrödinger equation, Numerical methods for hyperbolic and kinetic problems, IRMA Lect. Math. Theor. Phys. 7. Eur. Math. Soc., Zürich (2005) 131–141. [Google Scholar]
  27. E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time. Proc. Amer. Math. Soc. 126 (1998) 523–530. [CrossRef] [MathSciNet] [Google Scholar]
  28. S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21 (1999) 441–454. [CrossRef] [MathSciNet] [Google Scholar]
  29. C. Josserand and Y. Pomeau, Nonlinear aspects of the theory of Bose-Einstein condensates. Nonlinearity 14 (2001) R25–R62. [CrossRef] [Google Scholar]
  30. H. Li and C.-K. Lin, Semiclassical limit and well-posedness of nonlinear Schrödinger-Poisson systems. Electron. J. Diff. Eq. (2003) 17 (electronic). [Google Scholar]
  31. H. Liu and E. Tadmor, Semiclassical limit of the nonlinear Schrödinger-Poisson equation with subcritical initial data. Methods Appl. Anal. 9 (2002) 517–531. [MathSciNet] [Google Scholar]
  32. E. Madelung, Quanten theorie in Hydrodynamischer Form. Zeit. Physik 40 (1927) 322. [Google Scholar]
  33. T. Makino, S. Ukai and S. Kawashima, Sur la solution à support compact de l'équation d'Euler compressible. Japan J. Appl. Math. 3 (1986) 249–257. [CrossRef] [MathSciNet] [Google Scholar]
  34. P.A. Markowich, P. Pietra and C. Pohl, Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit. Numer. Math. 81 (1999) 595–630. [CrossRef] [MathSciNet] [Google Scholar]
  35. S. Masaki, Local existence and WKB approximation of solutions to Schrödinger-Poisson system in the two-dimensional whole space. Comm. Partial Differential Equations 35 (2010) 2253–2278. [CrossRef] [MathSciNet] [Google Scholar]
  36. V.P. Maslov and M.V. Fedoriuk, Semiclassical approximation in quantum mechanics, Mathematical Physics and Applied Mathematics 7. Translated from the Russian by J. Niederle and J. Tolar, Contemporary Mathematics 5. D. Reidel Publishing Co., Dordrecht (1981). [Google Scholar]
  37. G. Métivier, Remarks on the well-posedness of the nonlinear Cauchy problem, Geometric analysis of PDE and several complex variables, Contemp. Math. 368. Amer. Math. Soc., Providence, RI (2005) 337–356. [Google Scholar]
  38. H. Michinel, J. Campo-Táboas, R. García-Fernández, J.R. Salgueiro and M.L. Quiroga-Teixeiro, Liquid light condensates. Phys. Rev. E 65 (2002) 066604. [CrossRef] [Google Scholar]
  39. B. Mohammadi and J.H. Saiac, Pratique de la simulation numérique. Dunod, Paris (2003). [Google Scholar]
  40. J. Nocedal and S.J. Wright, Numerical optimization. 2d edition, Springer Series in Operations Research and Financial Engineering, Springer, New York (2006). [Google Scholar]
  41. L. Pitaevskii and S. Stringari, Bose-Einstein condensation, International Series of Monographs on Physics 116. The Clarendon Press Oxford University Press, Oxford (2003). [Google Scholar]
  42. E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energy–critical nonlinear Schrödinger equation in Formula . Amer. J. Math. 129 (2007) 1–60. [MathSciNet] [Google Scholar]
  43. G. Strang, Introduction to applied mathematics. Applied Mathematical Sciences, Wellesley-Cambridge Press, New York (1986). [Google Scholar]
  44. C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation, self-focusing and wave collapse. Springer-Verlag, New York (1999). [Google Scholar]
  45. M. Taylor, Partial differential equations. III, Applied Mathematical Sciences 117. Nonlinear equations. Springer-Verlag, New York (1997). [Google Scholar]
  46. L. Thomann, Instabilities for supercritical Schrödinger equations in analytic manifolds. J. Diff. Eq. 245 (2008) 249–280. [CrossRef] [Google Scholar]
  47. Z. Xin, Blowup of smooth solutions of the compressible Navier-Stokes equation with compact density. Comm. Pure Appl. Math. 51 (1998) 229–240. [CrossRef] [MathSciNet] [Google Scholar]
  48. V.E. Zakharov and S.V. Manakov, On the complete integrability of a nonlinear Schrödinger equation. Theor. Math. Phys. 19 (1974) 551–559. [CrossRef] [Google Scholar]
  49. V.E. Zakharov and A.B. Shabat, Interaction between solitons in a stable medium. Sov. Phys. JETP 37 (1973) 823–828. [Google Scholar]

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