The WKB method and geometric instability for nonlinear Schrödinger equations on surfaces
Bulletin de la Société Mathématique de France, Volume 136 (2008) no. 2, pp. 167-193.

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.

À l'aide de la méthode WKB nous construisons des solutions approchées à l'équation de Schrödinger cubique sur une variété qui possède une géodésique stable. Cette construction permet d'obtenir des résultats d'instabilités dans des espaces de Sobolev.

DOI: 10.24033/bsmf.2553
Classification: 35Q55, 35B35, 35R25
Keywords: nonlinear schrödinger equation, instability, quasimode
Mot clés : équation de schrödinger non linéaire, instabilité, quasi-mode 
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     author = {Thomann, Laurent},
     title = {The {WKB} method and geometric instability for nonlinear {Schr\"odinger} equations on surfaces},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
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     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {136},
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     doi = {10.24033/bsmf.2553},
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     url = {https://www.numdam.org/articles/10.24033/bsmf.2553/}
}
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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, Volume 136 (2008) no. 2, pp. 167-193. doi : 10.24033/bsmf.2553. https://www.numdam.org/articles/10.24033/bsmf.2553/

[1] S. Alinhac & P. Gérard - Opérateurs pseudo-différentiels et théorème de Nash-Moser, Savoirs Actuels, InterÉditions, Paris, 1991. | Zbl

[2] V. Banica - « On the nonlinear Schrödinger dynamics on 𝕊 2 », J. Math. Pures Appl. (9) 83 (2004), p. 77-98. | MR | Zbl

[3] J. Bourgain - « Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations », Geom. Funct. Anal. 3 (1993), p. 107-156. | MR | Zbl

[4] N. Burq, P. Gérad & N. Tzvetkov - « An instability property of the nonlinear Schrödinger equation on S d », Math. Res. Lett. 9 (2002), p. 323-335. | MR | Zbl

[5] N. Burq, P. Gérard & N. Tzvetkov - « Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds », Amer. J. Math. 126 (2004), p. 569-605. | MR | Zbl

[6] -, « Agmon estimates and nonlinear instability for Schrödinger equations », preprint, 2005.

[7] R. Carles - « Remarks on the Cauchy problem for nonlinear Schrödinger equations with potential », preprint arXiv:math.AP/0609391.

[8] -, « Geometric optics and instability for semi-classical Schrödinger equations », Arch. Ration. Mech. Anal. 183 (2007), p. 525-553. | MR | Zbl

[9] M. Christ, J. Colliander & T. Tao - « Ill-posedness for nonlinear Schrödinger and wave equation », to appear in Annales IHP.

[10] -, « Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations », Amer. J. Math. 125 (2003), p. 1235-1293. | MR | Zbl

[11] M. Combescure - « The quantum stability problem for some class of time-dependent Hamiltonians », Ann. Physics 185 (1988), p. 86-110. | MR | Zbl

[12] B. Helffer - Semi-classical analysis for the Schrödinger operator and applications, Lecture Notes in Mathematics, vol. 1336, Springer, 1988. | MR | Zbl

[13] W. Klingenberg - Riemannian geometry, de Gruyter Studies in Mathematics, vol. 1, Walter de Gruyter & Co., 1982. | MR | Zbl

[14] L. Perko - Differential equations and dynamical systems, Texts in Applied Mathematics, vol. 7, Springer, 1991. | MR | Zbl

[15] J. V. Ralston - « Approximate eigenfunctions of the Laplacian », J. Differential Geometry 12 (1977), p. 87-100. | MR | Zbl

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