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

Rémi Carles; Bijan Mohammadi

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • Volume: 45, Issue: 5, page 981-1008
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Carles, Rémi, and Mohammadi, Bijan. "Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime." ESAIM: Mathematical Modelling and Numerical Analysis 45.5 (2011): 981-1008. <http://eudml.org/doc/276351>.

@article{Carles2011,
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. },
author = {Carles, Rémi, Mohammadi, Bijan},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Nonlinear Schrödinger equation; semiclassical limit; compressible Euler equation; numerical simulation; nonlinear Schrödinger equation; compressible Euler equation; numerical examples; spectral method; Madelung transform; convergence; shocks},
language = {eng},
month = {6},
number = {5},
pages = {981-1008},
publisher = {EDP Sciences},
title = {Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime},
url = {http://eudml.org/doc/276351},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Carles, Rémi
AU - Mohammadi, Bijan
TI - Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/6//
PB - EDP Sciences
VL - 45
IS - 5
SP - 981
EP - 1008
AB - 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.
LA - eng
KW - Nonlinear Schrödinger equation; semiclassical limit; compressible Euler equation; numerical simulation; nonlinear Schrödinger equation; compressible Euler equation; numerical examples; spectral method; Madelung transform; convergence; shocks
UR - http://eudml.org/doc/276351
ER -

References

top
  1. F.Kh. Abdullaev, A. Gammal, L. Tomio and T. Frederico, Stability of trapped Bose-Einstein condensates. Phys. Rev. A63 (2001) 043604.  
  2. T. Alazard and R. Carles, Semi-classical limit of Schrödinger-Poisson equations in space dimension n 3 . J. Diff. Eq.233 (2007) 241–275.  Zbl1107.35018
  3. T. Alazard and R. Carles, Supercritical geometric optics for nonlinear Schrödinger equations. Arch. Rational Mech. Anal.194 (2009) 315–347.  Zbl1179.35302
  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 Linéaire26 (2009) 959–977.  Zbl1167.35328
  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.  Zbl1006.65112
  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.  Zbl1038.65099
  7. C. Besse, A relaxation scheme for the nonlinear Schrödinger equation. SIAM J. Numer. Anal.42 (2004) 934–952.  Zbl1077.65103
  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.  Zbl1026.65073
  9. Y. Brenier and L. Corrias, A kinetic formulation for multi-branch entropy solutions of scalar conservation laws. Ann. Inst. Henri Poincaré, Anal. Non Linéaire15 (1998) 169–190.  Zbl0893.35068
  10. R. Carles, Geometric optics and instability for semi-classical Schrödinger equations. Arch. Rational Mech. Anal.183 (2007) 525–553.  Zbl1134.35098
  11. R. Carles, Semi-classical analysis for nonlinear Schrödinger equations. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2008).  Zbl1153.35070
  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.  Zbl1162.35068
  13. T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics10. New York University Courant Institute of Mathematical Sciences, New York (2003).  Zbl1055.35003
  14. J.-Y. Chemin, Dynamique des gaz à masse totale finie. Asymptotic Anal.3 (1990) 215–220.  Zbl0708.76110
  15. D. Chiron and F. Rousset, Geometric optics and boundary layers for nonlinear Schrödinger equations. Comm. Math. Phys.288 (2009) 503–546.  Zbl1179.35303
  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.  
  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. Paris345 (2007) 531–536.  Zbl1128.65064
  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.  Zbl1176.35170
  19. J.J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math.27 (1974) 207–281.  Zbl0285.35010
  20. A. Gammal, T. Frederico, L. Tomio and Ph. Chomaz, Atomic Bose-Einstein condensation with three-body intercations and collective excitations. J. Phys. B33 (2000) 4053–4067.  
  21. C.L. Gardner, The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math.54 (1994) 409–427.  Zbl0815.35111
  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), www.numdam.org, pp. Exp. No. XIII, 13.  URIhttp://www.numdam.org/numdam-bin/fitem?id=SEDP_1992-1993____A13_0
  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.  Zbl0881.35099
  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.  Zbl0396.35028
  25. L. Gosse, Using K -branch entropy solutions for multivalued geometric optics computations. J. Comput. Phys.180 (2002) 155–182.  Zbl0999.78003
  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.  Zbl1210.81038
  27. E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time. Proc. Amer. Math. Soc.126 (1998) 523–530.  Zbl0910.35115
  28. S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput.21 (1999) 441–454.  Zbl0947.82008
  29. C. Josserand and Y. Pomeau, Nonlinear aspects of the theory of Bose-Einstein condensates. Nonlinearity14 (2001) R25–R62.  Zbl1037.82031
  30. H. Li and C.-K. Lin, Semiclassical limit and well-posedness of nonlinear Schrödinger-Poisson systems. Electron. J. Diff. Eq. (2003) 17 (electronic).  Zbl1055.35111
  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.  Zbl1166.35374
  32. E. Madelung, Quanten theorie in Hydrodynamischer Form. Zeit. Physik40 (1927) 322.  
  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.  Zbl0637.76065
  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.  Zbl0928.65109
  35. S. Masaki, Local existence and WKB approximation of solutions to Schrödinger-Poisson system in the two-dimensional whole space. Comm. Partial Differential Equations35 (2010) 2253–2278.  Zbl1232.35155
  36. V.P. Maslov and M.V. Fedoriuk, Semiclassical approximation in quantum mechanics, Mathematical Physics and Applied Mathematics7. Translated from the Russian by J. Niederle and J. Tolar, Contemporary Mathematics5. D. Reidel Publishing Co., Dordrecht (1981).  
  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.  Zbl1071.35074
  38. H. Michinel, J. Campo-Táboas, R. García-Fernández, J.R. Salgueiro and M.L. Quiroga-Teixeiro, Liquid light condensates. Phys. Rev. E65 (2002) 066604.  
  39. B. Mohammadi and J.H. Saiac, Pratique de la simulation numérique. Dunod, Paris (2003).  
  40. J. Nocedal and S.J. Wright, Numerical optimization. 2d edition, Springer Series in Operations Research and Financial Engineering, Springer, New York (2006).  Zbl1104.65059
  41. L. Pitaevskii and S. Stringari, Bose-Einstein condensation, International Series of Monographs on Physics116. The Clarendon Press Oxford University Press, Oxford (2003).  Zbl1110.82002
  42. E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energy–critical nonlinear Schrödinger equation in 1 + 4 . Amer. J. Math.129 (2007) 1–60.  Zbl1160.35067
  43. G. Strang, Introduction to applied mathematics. Applied Mathematical Sciences, Wellesley-Cambridge Press, New York (1986).  Zbl0618.00015
  44. C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation, self-focusing and wave collapse. Springer-Verlag, New York (1999).  Zbl0928.35157
  45. M. Taylor, Partial differential equations. III, Applied Mathematical Sciences117. Nonlinear equations. Springer-Verlag, New York (1997).  
  46. L. Thomann, Instabilities for supercritical Schrödinger equations in analytic manifolds. J. Diff. Eq.245 (2008) 249–280.  Zbl1157.35107
  47. Z. Xin, Blowup of smooth solutions of the compressible Navier-Stokes equation with compact density. Comm. Pure Appl. Math.51 (1998) 229–240.  Zbl0937.35134
  48. V.E. Zakharov and S.V. Manakov, On the complete integrability of a nonlinear Schrödinger equation. Theor. Math. Phys.19 (1974) 551–559.  Zbl0298.35016
  49. V.E. Zakharov and A.B. Shabat, Interaction between solitons in a stable medium. Sov. Phys. JETP37 (1973) 823–828.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.