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Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime

Published online by Cambridge University Press:  10 June 2011

Rémi Carles  and
Bijan Mohammadi
Rémi Carles
Affiliation:
CNRS and Univ. Montpellier 2, Mathématiques, CC 051, 34095 Montpellier, France. remi.carles@math.cnrs.fr
Bijan Mohammadi
Affiliation:
CNRS and Univ. Montpellier 2, Mathématiques, CC 051, 34095 Montpellier, France. remi.carles@math.cnrs.fr
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Abstract

We study numerically the semiclassical limit for the nonlinear Schrödinger equation thanks to a modification of the Madelung transform due to Grenier. This approach allows for the presence of vacuum. Even if the mesh size and the time step do not depend on the Planck constant, we recover the position and current densities in the semiclassical limit, with a numerical rate of convergence in accordance with the theoretical results, before shocks appear in the limiting Euler equation. By using simple projections, the mass and the momentum of the solution are well preserved by the numerical scheme, while the variation of the energy is not negligible numerically. Experiments suggest that beyond the critical time for the Euler equation, Grenier's approach yields smooth but highly oscillatory terms.

Keywords

Nonlinear Schrödinger equationsemiclassical limitcompressible Euler equationnumerical simulation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 5 , September 2011 , pp. 981 - 1008
Copyright
© EDP Sciences, SMAI, 2011

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References

Abdullaev, F.Kh., Gammal, A., Tomio, L. and Frederico, T., Stability of trapped Bose-Einstein condensates. Phys. Rev. A 63 (2001) 043604. CrossRef
Alazard, T. and Carles, R., Semi-classical limit of Schrödinger-Poisson equations in space dimension $n\ge 3$. J. Diff. Eq. 233 (2007) 241–275. CrossRef
Alazard, T. and Carles, R., Supercritical geometric optics for nonlinear Schrödinger equations. Arch. Rational Mech. Anal. 194 (2009) 315–347. CrossRef
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.
Bao, W., Jin, S. and Markowich, P.A., On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime. J. Comput. Phys. 175 (2002) 487–524. CrossRef
Bao, W., Jin, S. and Markowich, P.A., Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes. SIAM J. Sci. Comput. 25 (2003) 27–64. CrossRef
Besse, C., A relaxation scheme for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 42 (2004) 934–952. CrossRef
Besse, C., Bidégaray, B. and Descombes, S., Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40 (2002) 26–40. CrossRef
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.
Carles, R., Geometric optics and instability for semi-classical Schrödinger equations. Arch. Rational Mech. Anal. 183 (2007) 525–553. CrossRef
R. Carles, Semi-classical analysis for nonlinear Schrödinger equations. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2008).
Carles, R. and Gosse, L., Numerical aspects of nonlinear Schrödinger equations in the presence of caustics. Math. Models Methods Appl. Sci. 17 (2007) 1531–1553. CrossRef
T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics 10. New York University Courant Institute of Mathematical Sciences, New York (2003).
Chemin, J.-Y., Dynamique des gaz à masse totale finie. Asymptotic Anal. 3 (1990) 215–220.
Chiron, D. and Rousset, F., Geometric optics and boundary layers for nonlinear Schrödinger equations. Comm. Math. Phys. 288 (2009) 503–546. CrossRef
Dalfovo, F., Giorgini, S., Pitaevskii, L.P. and Stringari, S., Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys. 71 (1999) 463–512. CrossRef
Degond, P., Gallego, S. and Méhats, F., An asymptotic preserving scheme for the Schrödinger equation in the semiclassical limit. C.R. Math. Acad. Sci. Paris 345 (2007) 531–536. CrossRef
Degond, P., Jin, S. and Tang, M., 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
Duistermaat, J.J., Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27 (1974) 207–281. CrossRef
Gammal, A., Frederico, T., Tomio, L. and Chomaz, Ph., Atomic Bose-Einstein condensation with three-body intercations and collective excitations. J. Phys. B 33 (2000) 4053–4067. CrossRef
Gardner, C.L., The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54 (1994) 409–427. CrossRef
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), http://www.numdam.org/numdam-bin/fitem?id=SEDP_1992-1993____A13_0www.numdam.org, pp. Exp. No. XIII, 13.
Gérard, P., Markowich, P.A., Mauser, N.J. and Poupaud, F., Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50 (1997) 323–379. 3.0.CO;2-C>CrossRef
Ginibre, J. and Velo, G., On a class of nonlinear Schrödinger equations. I The Cauchy problem, general case. J. Funct. Anal. 32 (1979) 1–32. CrossRef
Gosse, L., Using $K$-branch entropy solutions for multivalued geometric optics computations. J. Comput. Phys. 180 (2002) 155–182. CrossRef
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.
Grenier, E., Semiclassical limit of the nonlinear Schrödinger equation in small time. Proc. Amer. Math. Soc. 126 (1998) 523–530. CrossRef
Jin, S., Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21 (1999) 441–454. CrossRef
Josserand, C. and Pomeau, Y., Nonlinear aspects of the theory of Bose-Einstein condensates. Nonlinearity 14 (2001) R25–R62. CrossRef
H. Li and C.-K. Lin, Semiclassical limit and well-posedness of nonlinear Schrödinger-Poisson systems. Electron. J. Diff. Eq. (2003) 17 (electronic).
Liu, H. and Tadmor, E., Semiclassical limit of the nonlinear Schrödinger-Poisson equation with subcritical initial data. Methods Appl. Anal. 9 (2002) 517–531.
Madelung, E., Quanten theorie in Hydrodynamischer Form. Zeit. Physik 40 (1927) 322. CrossRef
Makino, T., Ukai, S. and Kawashima, S., Sur la solution à support compact de l'équation d'Euler compressible. Japan J. Appl. Math. 3 (1986) 249–257. CrossRef
Markowich, P.A., Pietra, P. and Pohl, C., Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit. Numer. Math. 81 (1999) 595–630. CrossRef
Masaki, S., 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
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).
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.
Michinel, H., Campo-Táboas, J., García-Fernández, R., Salgueiro, J.R. and Quiroga-Teixeiro, M.L., Liquid light condensates. Phys. Rev. E 65 (2002) 066604. CrossRef
B. Mohammadi and J.H. Saiac, Pratique de la simulation numérique. Dunod, Paris (2003).
J. Nocedal and S.J. Wright, Numerical optimization. 2d edition, Springer Series in Operations Research and Financial Engineering, Springer, New York (2006).
L. Pitaevskii and S. Stringari, Bose-Einstein condensation, International Series of Monographs on Physics 116. The Clarendon Press Oxford University Press, Oxford (2003).
Ryckman, E. and Visan, M., Global well-posedness and scattering for the defocusing energy–critical nonlinear Schrödinger equation in ${\mathbb R}^{1+4}$. Amer. J. Math. 129 (2007) 1–60. CrossRef
G. Strang, Introduction to applied mathematics. Applied Mathematical Sciences, Wellesley-Cambridge Press, New York (1986).
C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation, self-focusing and wave collapse. Springer-Verlag, New York (1999).
M. Taylor, Partial differential equations. III, Applied Mathematical Sciences 117. Nonlinear equations. Springer-Verlag, New York (1997).
Thomann, L., Instabilities for supercritical Schrödinger equations in analytic manifolds. J. Diff. Eq. 245 (2008) 249–280. CrossRef
Xin, Z., Blowup of smooth solutions of the compressible Navier-Stokes equation with compact density. Comm. Pure Appl. Math. 51 (1998) 229–240. 3.0.CO;2-C>CrossRef
Zakharov, V.E. and Manakov, S.V., On the complete integrability of a nonlinear Schrödinger equation. Theor. Math. Phys. 19 (1974) 551–559. CrossRef
Zakharov, V.E. and Shabat, A.B., Interaction between solitons in a stable medium. Sov. Phys. JETP 37 (1973) 823–828.