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

 Access to full text

 Full (PDF)

1. [1] S. Alinhac & P. Gérard – Opérateurs pseudo-différentiels et théorème de Nash-Moser, Savoirs Actuels, InterÉditions, Paris, 1991. Zbl0791.47044
2. [2] V. Banica – « On the nonlinear Schrödinger dynamics on ${𝕊}^2$ », J. Math. Pures Appl. (9) 83 (2004), p. 77–98. Zbl1084.35082MR2032582
3. [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. Zbl0787.35097MR1209299
4. [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. Zbl1003.35113MR1909648
5. [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. Zbl1067.58027MR2058384
6. [6] —, « Agmon estimates and nonlinear instability for Schrödinger equations », preprint, 2005. 
7. [7] R. Carles – « Remarks on the Cauchy problem for nonlinear Schrödinger equations with potential », preprint arXiv:math.AP/0609391. 
8. [8] —, « Geometric optics and instability for semi-classical Schrödinger equations », Arch. Ration. Mech. Anal.183 (2007), p. 525–553. Zbl1134.35098MR2278414
9. [9] M. Christ, J. Colliander & T. Tao – « Ill-posedness for nonlinear Schrödinger and wave equation », to appear in Annales IHP. Zbl1048.35101
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
15. [15] J. V. Ralston – « Approximate eigenfunctions of the Laplacian », J. Differential Geometry12 (1977), p. 87–100. Zbl0385.58012MR470998

1. Sebastian Herr, Energy Critical nonlinear Schrödinger equations in the presence of periodic geodesics