The WKB method and geometric instability for nonlinear Schrödinger equations on surfaces

Laurent Thomann

Bulletin de la Société Mathématique de France (2008)

  • Volume: 136, Issue: 2, page 167-193
  • ISSN: 0037-9484

Abstract

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In this paper we are interested in constructing WKB approximations for the nonlinear cubic Schrödinger equation on a Riemannian surface which has a stable geodesic. These approximate solutions will lead to some instability properties of the equation.

How to cite

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Thomann, Laurent. "The WKB method and geometric instability for nonlinear Schrödinger equations on surfaces." Bulletin de la Société Mathématique de France 136.2 (2008): 167-193. <http://eudml.org/doc/272410>.

@article{Thomann2008,
abstract = {In this paper we are interested in constructing WKB approximations for the nonlinear cubic Schrödinger equation on a Riemannian surface which has a stable geodesic. These approximate solutions will lead to some instability properties of the equation.},
author = {Thomann, Laurent},
journal = {Bulletin de la Société Mathématique de France},
keywords = {nonlinear schrödinger equation; instability; quasimode},
language = {eng},
number = {2},
pages = {167-193},
publisher = {Société mathématique de France},
title = {The WKB method and geometric instability for nonlinear Schrödinger equations on surfaces},
url = {http://eudml.org/doc/272410},
volume = {136},
year = {2008},
}

TY - JOUR
AU - Thomann, Laurent
TI - The WKB method and geometric instability for nonlinear Schrödinger equations on surfaces
JO - Bulletin de la Société Mathématique de France
PY - 2008
PB - Société mathématique de France
VL - 136
IS - 2
SP - 167
EP - 193
AB - In this paper we are interested in constructing WKB approximations for the nonlinear cubic Schrödinger equation on a Riemannian surface which has a stable geodesic. These approximate solutions will lead to some instability properties of the equation.
LA - eng
KW - nonlinear schrödinger equation; instability; quasimode
UR - http://eudml.org/doc/272410
ER -

References

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  10. [10] —, « Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations », Amer. J. Math.125 (2003), p. 1235–1293. Zbl1048.35101MR2018661
  11. [11] M. Combescure – « The quantum stability problem for some class of time-dependent Hamiltonians », Ann. Physics185 (1988), p. 86–110. Zbl0655.35076MR954669
  12. [12] B. Helffer – Semi-classical analysis for the Schrödinger operator and applications, Lecture Notes in Mathematics, vol. 1336, Springer, 1988. Zbl0647.35002MR960278
  13. [13] W. Klingenberg – Riemannian geometry, de Gruyter Studies in Mathematics, vol. 1, Walter de Gruyter & Co., 1982. Zbl0495.53036MR666697
  14. [14] L. Perko – Differential equations and dynamical systems, Texts in Applied Mathematics, vol. 7, Springer, 1991. Zbl0717.34001MR1083151
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