84
Views
0
CrossRef citations to date
0
Altmetric
Research Article

A weakly turbulent solution to the cubic nonlinear harmonic oscillator on ℝ2 perturbed by a real smooth potential decaying to zero at infinity

Pages 185-216 | Received 20 Jun 2023, Accepted 01 Jan 2024, Published online: 27 Jan 2024

References

  • Bourgain, J. (2010). Problems in Hamiltonian PDE’s. In: Alon, N., Bourgain, J., Connes, A., Gromov, M., Milman, V., eds. Visions in Mathematics: GAFA 2000 Special Volume, Part I. Basel: Birkhäuser, 32–56.
  • Kuksin, S. B. (1997). Oscillations in space-periodic nonlinear Schrödinger equations. Geom. Funct. Anal. GAFA 7(2):338–363. DOI: 10.1007/PL00001622.
  • Kuksin, S. B. (1995). On squeezing and flow of energy for nonlinear wave equations. Geom. Funct. Anal. (GAFA) 5:668–701. DOI: 10.1007/BF01902057.
  • Kuksin, S. B. (1996). Growth and oscillations of solutions of nonlinear Schrödinger equation. Commun. Math. Phys. 178(2):265–280. DOI: 10.1007/BF02099448.
  • Kuksin, S. B. (1997). On turbulence in nonlinear Schrödinger equations. Geom. Funct. Anal. 7(4):783–822. DOI: 10.1007/s000390050026.
  • Kuksin, S. B. (1999). Spectral properties of solutions for nonlinear PDEs in the turbulent regime. Geomet. Funct. Anal. GAFA 9(1):141–184. DOI: 10.1007/s000390050083.
  • Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T. (2010). Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation. Invent. Math. 181(1):39–113. DOI: 10.1007/s00222-010-0242-2.
  • Guardia, M., Kaloshin, V. (2015). Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation. J. Eur. Math. Soc. 17(1):71–149. DOI: 10.4171/jems/499.
  • Haus, E., Procesi, M. (2015). Growth of Sobolev norms for the quintic NLS on T2. Anal. PDE 8(4):883–922. DOI: 10.2140/apde.2015.8.883.
  • Guardia, M., Haus, E., Procesi, M. (2016). Growth of Sobolev norms for the analytic NLS on T2. Adv. Math. 301:615–692. DOI: 10.1016/j.aim.2016.06.018.
  • Giuliani, F., Guardia, M. (2022). Sobolev norms explosion for the cubic NLS on irrational tori. Nonlinear Anal. 220:112865. DOI: 10.1016/j.na.2022.112865.
  • Hani, Z., Pausader, B., Tzvetkov, N., Visciglia, N. (2015). Modified scattering for the cubic Schrödinger equation on product spaces and applications. Forum Math. Pi 3:e4. DOI: 10.1017/fmp.2015.5.
  • Hani, Z. (2014). Long-time instability and unbounded Sobolev orbits for some periodic nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 211:929–964. DOI: 10.1007/s00205-013-0689-6.
  • Guardia, M., et al. (2022). Strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite gap tori for the 2D cubic NLS equation. J. Eur. Math. Soc. 25(4):1497–1551. DOI: 10.4171/JEMS/1200.
  • Grellier, S., Gerard, P. (2015). The cubic Szego equation and Hankel operators. arXiv preprint arXiv:1508.06814.
  • Gérard, P., Lenzmann, E. (2022). The Calogero-Moser derivative nonlinear Schrödinger equation. arXiv preprint arXiv:2208.04105.
  • Staffilani, G. (1997). On the growth of high Sobolev norms of solutions for K d V and Schrödinger equations. Duke Math. J. 86(1):109–142 DOI: 10.1215/S0012-7094-97-08604-X.
  • Colliander, J., Delort, J.-M., Kenig, C. E., Staffilani, G. (2001). Bilinear estimates and applications to 2D NLS. Trans. Amer. Math. Soc. 353(8):3307–3325. DOI: 10.1090/S0002-9947-01-02760-X.
  • Zhong, S. (2008). The growth in time of higher Sobolev norms of solutions to Schrödinger equations on compact Riemannian manifolds. J. Differ. Equ. 245(2):359–376. DOI: 10.1016/j.jde.2008.03.008.
  • Catoire, F., Wang, W.-M. (2008). Bounds on Sobolev norms for the nonlinear Schrödinger equation on general tori. arXiv preprint arXiv:0809.4633.
  • Sohinger, V. (2011). Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on Sˆ1. Differ. Integral Equ. 24(7/8):653–718.
  • Sohinger, V. (2011). Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on R. Indiana Univ. Math. J. 60:1487–1516. DOI: 10.1512/iumj.2011.60.4399.
  • Colliander, J., Kwon, S., Oh, T. (2012). A remark on normal forms and the “upside-down” I-method for periodic NLS: growth of higher Sobolev norms. J. d’Analyse Math. 118(1):55–82. DOI: 10.1007/s11854-012-0029-z.
  • Planchon, F., Tzvetkov, N., Visciglia, N. (2017). On the growth of Sobolev norms for NLS on 2-and 3-dimensional manifolds. Anal. PDE 10(5):1123–1147. DOI: 10.2140/apde.2017.10.1123.
  • Planchon, F., Tzvetkov, N., Visciglia, N. (2022). Growth of Sobolev Norms for 2d NLS with harmonic potential. Revista Matemática Iberoamericana 39(4):1405–1436. DOI: 10.4171/RMI/1371.
  • Faou, E., Raphaël, P. (2020). On weakly turbulent solutions to the perturbed linear harmonic oscillator. arXiv preprint arXiv:2006.08206.
  • Bourgain, J. (1999). Growth of Sobolev norms in linear Schrödinger equations with quasi-periodic potential. Commun. Math. Phys. 204:207–247. DOI: 10.1007/s002200050644.
  • Bourgain, J. (1999). On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential. J. d’Analyse Math. 77(1):315–348. DOI: 10.1007/BF02791265.
  • Maspero, A., Robert, D. (2017). On time dependent Schrödinger equations: Global well-posedness and growth of Sobolev norms. J. Funct. Anal. 273(2):721–781. DOI: 10.1016/j.jfa.2017.02.029.
  • Bambusi, D., Grébert, B., Maspero, A., Robert, D. (2021). Growth of Sobolev norms for abstract linear Schrödinger equations. J. Eur. Math. Soc. 23(2):557–583. DOI: 10.4171/jems/1017.
  • Bambusi, D., Langella, B., Montalto, R. (2022). Growth of Sobolev norms for unbounded perturbations of the Schrödinger equation on flat tori. J. Differ. Equ. 318:344–358. DOI: 10.1016/j.jde.2022.02.024.
  • Bambusi, D., Langella, B. (2022). Growth of Sobolev norms in quasi integrable quantum systems. arXiv preprint arXiv:2202.04505.
  • Maspero, A. (2022). Growth of Sobolev norms in linear Schrödinger equations as a dispersive phenomenon. Adv. Math. 411:108800. DOI: 10.1016/j.aim.2022.108800.
  • Maspero, A. (2023). Generic transporters for the linear time-dependent quantum harmonic oscillator on R. Int. Math. Res. Not. 2023(14):12088–12118. DOI: 10.1093/imrn/rnac174.
  • Burak Erdoğan, M., Killip, R., Schlag, W. (2003). Energy growth in Schrödinger’s equation with Markovian forcing. Commun. Math. Phys. 240(1–2):1–29. DOI: 10.1007/s00220-003-0892-7.
  • Eliasson, H. L., Kuksin, S. B. (2009). On reducibility of Schrödinger equations with quasiperiodic in time potentials. In: Commun. Math. Phys. 286:125–135. DOI: 10.1007/s00220-008-0683-2.
  • Delort, J.-M. (2010). Growth of Sobolev norms of solutions of linear Schrödinger equations on some compact manifolds. Int. Math. Res. Not. 2010(12):2305–2328.
  • Wang, W.-M. (2008). Logarithmic bounds on Sobolev norms for time dependent linear Schrödinger equations. Commun. Partial Differ. Equ. 33(12):2164–2179. DOI: 10.1080/03605300802537115.
  • Arnol’d, V. I. (2020). Instability of dynamical systems with several degrees of freedom. In: MacKay, R. S., Meiss, J. D., eds. Hamiltonian Dynamical Systems. Boca Raton, FL: CRC Press, 633–637.
  • Delshams, A., Gidea, M., de la Llave, R., Seara, T. M. (2008). Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation. In: Craig, W., ed. Hamiltonian Dynamical Systems and Applications. Dordrecht: Springer, 285–336.
  • Chabert, M. (2023). Weakly turbulent solution to Schrödinger equation on the two-dimensional torus with real potential decaying at infinity. arXiv preprint arXiv:2305.15939.
  • Germain, P., Hani, Z., Thomann, L. (2016). On the continuous resonant equation for NLS. I. Deterministic analysis. Journal de Mathématiques Pures et Appliquées 105(1):131–163. DOI: 10.1016/j.matpur.2015.10.002.
  • Kato, T. (2013). Perturbation Theory for Linear Operators, Vol. 132. Springer.
  • Kavian, O., Weissler, F. B. (1994). Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation. Michigan Math. J. 41(1):151–173. DOI: 10.1307/mmj/1029004922.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.