High frequency solutions of the nonlinear Schrödinger equation on surfaces

Burq, Nicolas; Gérard, Patrick; Tzvetkov, Nikolay

doi:10.1090/S0033-569X-09-01178-1

Authors: Nicolas Burq, Patrick Gérard and Nikolay Tzvetkov
Journal: Quart. Appl. Math. 68 (2010), 61-71
MSC (2000): Primary 35Q55; Secondary 35B30
DOI: https://doi.org/10.1090/S0033-569X-09-01178-1
Published electronically: October 20, 2009
MathSciNet review: 2598880
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We address the problem of describing solutions of the nonlinear Schrödin-ger equation on a compact surface in the high frequency regime. In this context, we introduce a nonnegative threshold, depending on the geometry of the surface, which can be seen as a measurement of the nonlinear character of the equation, and we compute this number for the torus and for the sphere, as a consequence of earlier arguments. The last part is devoted to the study, on the sphere, of the critical regime associated to this threshold. We prove that the effective dynamics are described by a new evolution equation, the Resonant Hermite-Schrödinger equation.

References

Naoufel Ben Abdallah, Florian Méhats, Christian Schmeiser, and Rada M. Weishäupl, The nonlinear Schrödinger equation with a strongly anisotropic harmonic potential, SIAM J. Math. Anal. 37 (2005), no. 1, 189–199. MR 2176928, DOI https://doi.org/10.1137/040614554
Björn Birnir, Carlos E. Kenig, Gustavo Ponce, Nils Svanstedt, and Luis Vega, On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations, J. London Math. Soc. (2) 53 (1996), no. 3, 551–559. MR 1396718, DOI https://doi.org/10.1112/jlms/53.3.551
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107–156. MR 1209299, DOI https://doi.org/10.1007/BF01896020
H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal. 4 (1980), no. 4, 677–681. MR 582536, DOI https://doi.org/10.1016/0362-546X%2880%2990068-1
N. Burq, P. Gérard, and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (2004), no. 3, 569–605. MR 2058384
N. Burq, P. Gérard, and N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on $S^d$, Math. Res. Lett. 9 (2002), no. 2-3, 323–335. MR 1909648, DOI https://doi.org/10.4310/MRL.2002.v9.n3.a8
N. Burq, P. Gérard, and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math. 159 (2005), no. 1, 187–223 (English, with English and French summaries). MR 2142336, DOI https://doi.org/10.1007/s00222-004-0388-x
Nicolas Burq, Patrick Gérard, and Nikolay Tzvetkov, Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 2, 255–301 (English, with English and French summaries). MR 2144988, DOI https://doi.org/10.1016/j.ansens.2004.11.003
N. Burq, P. Gérard, and N. Tzvetkov, The Cauchy problem for the nonlinear Schrödinger equation on compact manifolds, Phase space analysis of partial differential equations. Vol. I, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2004, pp. 21–52. MR 2144405

Catoire, F., Wang, W.-M., Bounds on Sobolev norms for the nonlinear Schrödinger equation on general tori. Preprint, September 2008, http://arxiv.org/abs/0809.4633.

B. Dehman, P. Gérard, and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z. 254 (2006), no. 4, 729–749. MR 2253466, DOI https://doi.org/10.1007/s00209-006-0005-3

László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math. 167 (2007), no. 3, 515–614. MR 2276262, DOI https://doi.org/10.1007/s00222-006-0022-1

P. Gérard, Nonlinear Schrödinger equations on compact manifolds, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, pp. 121–139. MR 2185741

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 4, 309–327 (English, with French summary). MR 801582

Bernard Helffer, Semi-classical analysis for the Schrödinger operator and applications, Lecture Notes in Mathematics, vol. 1336, Springer-Verlag, Berlin, 1988. MR 960278

Carlos E. Kenig, Gustavo Ponce, and Luis Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (2001), no. 3, 617–633. MR 1813239, DOI https://doi.org/10.1215/S0012-7094-01-10638-8

Takayoshi Ogawa, A proof of Trudinger’s inequality and its application to nonlinear Schrödinger equations, Nonlinear Anal. 14 (1990), no. 9, 765–769. MR 1049119, DOI https://doi.org/10.1016/0362-546X%2890%2990104-O

Christopher D. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J. 53 (1986), no. 1, 43–65. MR 835795, DOI https://doi.org/10.1215/S0012-7094-86-05303-2

Gigliola Staffilani and Daniel Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations 27 (2002), no. 7-8, 1337–1372. MR 1924470, DOI https://doi.org/10.1081/PDE-120005841

Catherine Sulem and Pierre-Louis Sulem, The nonlinear Schrödinger equation, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999. Self-focusing and wave collapse. MR 1696311

Laurent Thomann, The WKB method and geometric instability for nonlinear Schrödinger equations on surfaces, Bull. Soc. Math. France 136 (2008), no. 2, 167–193 (English, with English and French summaries). MR 2415340, DOI https://doi.org/10.24033/bsmf.2553

M. V. Vladimirov, On the solvability of a mixed problem for a nonlinear equation of Schrödinger type, Dokl. Akad. Nauk SSSR 275 (1984), no. 4, 780–783 (Russian). MR 745511