## Energy supercritical nonlinear Schrödinger equations: quasiperiodic solutions.(English)Zbl 1342.35358

Summary: We construct time quasiperiodic solutions to the energy supercritical nonlinear Schrödinger equations on the torus in arbitrary dimensions. This introduces a new approach, which could have general applicability.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35M10 PDEs of mixed type 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series 35B10 Periodic solutions to PDEs 35B32 Bifurcations in context of PDEs
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### References:

 [1] J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations , Ann. of Math. (2) 148 (1998), 363-439. · Zbl 0928.35161 [2] J. Bourgain, “Nonlinear Schrödinger equations” in Hyperbolic Equations and Frequency Interactions (Park City, Utah, 1995) , IAS/Park City Math. Ser. 5 , Amer. Math. Soc., Providence, 1999, 3-157. · Zbl 0952.35127 [3] J. Bourgain, Green’s Function Estimates for Latttice Schrödinger Operators and Applications , Ann. of Math. Stud. 158 (2005), Princeton Univ. Press, Princeton. · Zbl 1137.35001 [4] J. Bourgain and W.-M. Wang, Quasi-periodic solutions of nonlinear random Schrödinger equations , J. Eur. Math. Soc. (JEMS) 10 (2008), 1-45. · Zbl 1148.35104 [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation , Invent. Math. 181 (2010), 39-113. · Zbl 1197.35265 [6] W. Craig and C. E. Wayne, Newton’s method and periodic solutions of nonlinear wave equations , Comm. Pure Appl. Math. 46 (1993), 1409-1498. · Zbl 0794.35104 [7] L. H. Eliasson and S. B. Kuksin, KAM for the nonlinear Schrödinger equation , Ann. of Math. (2) 172 (2010), 371-435. · Zbl 1201.35177 [8] J. Fröhlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy , Comm. Math. Phys. 88 (1983), 151-184. · Zbl 0519.60066 [9] J. Geng, X. Xu, and J. You, An infinite-dimensional KAM theorem and its application to the two-dimensional cubic Schrödinger equation , Adv. Math. 266 (2011), 5361-5402. · Zbl 1213.37104 [10] M. Goldstein and W. Schlag, Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions , Ann. of Math. (2) 154 (2001), 155-203. · Zbl 0990.39014 [11] M. Guardia and V. Kaloshin, Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation , J. Eur. Math. Soc. (JEMS) 17 (2015), 71-149. · Zbl 1311.35284 [12] S. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic osillations for a nonlinear Schrödinger equation , Ann. of Math. (2) 143 (1996), 149-179. · Zbl 0847.35130 [13] M. Procesi and C. Procesi, A normal form for the Schrödinger equation with analytic non-linearities , Comm. Math. Phys. 312 (2012), 501-557. · Zbl 1277.35318 [14] M. Procesi and C. Procesi, A KAM algorithm for the resonant non-linear Schrödinger equation , Adv. Math. 272 (2015), 399-470. · Zbl 1312.37047 [15] J. Schur, Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, I , J. Reine Angew. Math. 147 (1917), 205-232. [16] J. Schur, Uber Potenzreihen, die im Innern des Einheitskreises beschränkt sind, II , J. Reine Angew. Math. 148 (1918), 122-145. [17] W.-M. Wang, Bounded Sobolev norms for linear Schrödinger equations under resonant perturbations , J. Funct. Anal. 254 (2008), 2926-2946. · Zbl 1171.35029 [18] W.-M. Wang, Pure point spectrum of the Floquet Hamiltonian for the quantum harmonic oscillator under time quasi-periodic perturbations , Comm. Math. Phys. 277 (2008), 459-496. · Zbl 1144.81018 [19] W.-M. Wang, Eigenfunction localization for the 2D periodic Schrödinger operator , Int. Math. Res. Not. (IMRN) 2011 , 1804-1838. · Zbl 1229.35154 [20] W.-M. Wang, Quasi-periodic solutions for nonlinear wave equations , C. R. Math. Acad. Sci. Paris 353 (2015), 601-604. · Zbl 1326.35203 [21] W.-M. Wang, Quasi-periodic solutions for nonlinear wave equations , preprint, 2015. · Zbl 1326.35203
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