×

KAM for beating solutions of the quintic NLS. (English) Zbl 1375.35488

Authors’ abstract: We consider the nonlinear Schrödinger equation of degree 5 on the circle. We prove the existence of quasi-periodic solutions with four frequencies which bifurcate from “resonant” solutions (studied in [B. Grébert and L. Thomann, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 29, No. 3, 455–477 (2012; Zbl 1259.37045)]) of the system obtained by truncating the Hamiltonian after one step of Birkhoff normal form, exhibiting recurrent exchange of energy between some Fourier modes. The existence of these quasi-periodic solutions is a purely nonlinear effect.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)

Citations:

Zbl 1259.37045
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bambusi D.: Birkhoff normal form for some nonlinear PDEs. Commun. Math. Phys. 234(2), 253-285 (2003) · Zbl 1032.37051 · doi:10.1007/s00220-002-0774-4
[2] Bambusi D., Berti M.: A Birkhoff-Lewis type theorem for some Hamiltonian PDEs. SIAM J. Math. Anal. 37(1), 83-102 (2005) · Zbl 1105.37045 · doi:10.1137/S0036141003436107
[3] Berti M., Biasco L., Procesi M.: KAM theory for the Hamiltonian derivative wave equation. Ann. Sci. Éc. Norm. Supér. 46(2), 301-373 (2013) · Zbl 1304.37055 · doi:10.24033/asens.2190
[4] Biasco L., Chierchia L.: On the measure of Lagrangian invariant tori in nearly-integrable mechanical systems. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26(4), 423-432 (2015) · Zbl 1361.37059 · doi:10.4171/RLM/713
[5] Biasco L., Di Gregorio L.: A Birkhoff-Lewis type theorem for the nonlinear wave equation. Arch. Ration. Mech. Anal. 196(1), 303-362 (2010) · Zbl 1197.35169 · doi:10.1007/s00205-009-0240-y
[6] Bourgain J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3(2), 107-156 (1993) · Zbl 0787.35097 · doi:10.1007/BF01896020
[7] Bourgain J.: Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE. Intern. Math. Res. Not. 11, 475-497 (1994) · Zbl 0817.35102 · doi:10.1155/S1073792894000516
[8] Bourgain J.: Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations. Geom. Funct. Anal. 6(2), 201-230 (1996) · Zbl 0872.35007 · doi:10.1007/BF02247885
[9] Bourgain J.: On diffusion in high-dimensional Hamiltonian systems and PDE. J. Anal. Math. 80, 1-35 (2000) · Zbl 0964.35143 · doi:10.1007/BF02791532
[10] Chierchia L., You J.: KAM tori for 1D nonlinear wave equations with periodic boundary conditions. Commun. Math. Phys. 211, 497-525 (2000) · Zbl 0956.37054 · doi:10.1007/s002200050824
[11] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T.: Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation. Invent. Math. 181(1), 39-113 (2010) · Zbl 1197.35265 · doi:10.1007/s00222-010-0242-2
[12] Corsi, L., Feola, R., Procesi, M.: Finite-dimensional invariant tori for tame vector fields. arXiv:1603.01204 (2016) · Zbl 1420.37108
[13] Craig W., Wayne C.E.: Newton’s method and periodic solutions of nonlinear wave equation. Commun. Pure Appl. Math. 46, 1409-1498 (1993) · Zbl 0794.35104 · doi:10.1002/cpa.3160461102
[14] Eliasson L.H.: Almost reducibility of linear quasi-periodic systems. Proc. Symp. Pure Math 69, 679-705 (2001) · Zbl 1015.34028 · doi:10.1090/pspum/069/1858550
[15] Eliasson H., Kuksin S.: KAM for the nonlinear Schrödinger equation. Ann. Math. 172(1), 371-435 (2010) · Zbl 1201.35177 · doi:10.4007/annals.2010.172.371
[16] Geng J., Yi Y.: Quasi-periodic solutions in a nonlinear Schrödinger equation. J. Diff. Equ. 233(2), 512-542 (2007) · Zbl 1108.37049 · doi:10.1016/j.jde.2006.07.027
[17] Grébert B., Paturel E., Thomann L.: Beating effects in cubic Schrödinger systems and growth of Sobolev norms. Nonlinearity 26, 1361-1376 (2013) · Zbl 1480.35352 · doi:10.1088/0951-7715/26/5/1361
[18] Grébert B., Thomann L.: Resonant dynamics for the quintic nonlinear Schrödinger equation. Ann. Inst. Henri Poincaré Anal. Non Linéaire 29(3), 455-477 (2012) · Zbl 1259.37045 · doi:10.1016/j.anihpc.2012.01.005
[19] Guardia M., Haus E., Procesi M.: Growth of Sobolev norms for the defocusing analytic NLS on T2. Adv. Math. 301(1), 615-692 (2016) · Zbl 1353.35260 · doi:10.1016/j.aim.2016.06.018
[20] Guardia M., Kaloshin V.: Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation. J. Eur. Math. Soc. (JEMS) 17, 71-149 (2015) · Zbl 1311.35284 · doi:10.4171/JEMS/499
[21] Hani, Z., Pausader, B., Tzvetkov, N., Visciglia, N.: Modified scattering for the cubic Schrödinger equation on product spaces and applications. Forum Math. Pi 3, e4 (2015) · Zbl 1326.35348
[22] Haus E., Procesi M.: Growth of Sobolev norms for the quintic NLS on T2. Anal. PDE 8, 883-922 (2015) · Zbl 1322.35126 · doi:10.2140/apde.2015.8.883
[23] Haus E., Thomann L.: Dynamics on resonant clusters for the quintic nonlinear Schrödinger equation. Dyn. Partial Differ. Equ. 10(2), 157-169 (2013) · Zbl 1346.35187 · doi:10.4310/DPDE.2013.v10.n2.a2
[24] Kuksin, S.: Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum. Funks. Anal. i Prilozhen 21:22-37, 95 (1987) · Zbl 0631.34069
[25] Kuksin S.B., Pöschel J.: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. Ann. Math. 143(1), 149-179 (1996) · Zbl 0847.35130 · doi:10.2307/2118656
[26] Liang Z., You J.: Quasi-periodic solutions for 1D Schrödinger equation with higher order nonlinearity SIAM. J. Math. Anal. 36, 1965-1990 (2005) · Zbl 1093.37030
[27] Procesi C., Procesi M.: A normal form for the Schrödinger equation with analytic non-linearities. Commun. Math. Phys. 312(2), 501-557 (2012) · Zbl 1277.35318 · doi:10.1007/s00220-012-1483-2
[28] Procesi C., Procesi M.: A KAM algorithm for the resonant non-linear Schrödinger equation. Adv. Math. 272, 399-470 (2015) · Zbl 1312.37047 · doi:10.1016/j.aim.2014.12.004
[29] Procesi C., Procesi M.: Reducible quasi-periodic solutions for the non linear Schrödinger equation. Boll. Unione Mat. Ital. 9(2), 189-236 (2016) · Zbl 1339.37077 · doi:10.1007/s40574-016-0066-0
[30] Procesi M., Procesi C., Van Nguyen B.: The energy graph of the non-linear Schrödinger equation. Rend. Lincei Mat. Appl. 24, 1-73 (2013) · Zbl 1294.35146
[31] Procesi M., Xu X.: Quasi-Töplitz Functions in KAM Theorem. SIAM J. Math. Anal. 45(4), 2148-2181 (2013) · Zbl 1304.37056 · doi:10.1137/110833014
[32] Van Nguyen, B.: Characteristic polynomials of the color marked graphs, related to the normal form of the nonlinear Schrödinger equation. arXiv:1203.6015 (2012) · Zbl 0708.35087
[33] Wayne E.: Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Commun. Math. Phys. 127, 479-528 (1990) · Zbl 0708.35087 · doi:10.1007/BF02104499
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.