Taiwanese Journal of Mathematics

A Nekhoroshev Type Theorem of Higher Dimensional Nonlinear Schrödinger Equations

Shidi Zhou and Jiansheng Geng

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In this paper, we prove a Nekhoroshev type theorem for high dimensional NLS (nonlinear Schrödinger equations):\[  \mathrm{i} \partial_{t} u - \Delta u + V * u  + \partial_{\overline{u}} g(x,u,\overline{u}) = 0, \quad    x \in \mathbb{T}^d, \; t \in \mathbb{R} \] where real-valued function $V$ is sufficiently smooth and $g$ is an analytic function. We prove that, for any given $M \in \mathbb{N}$, there exists an $\varepsilon_0 \gt 0$, such that for any solution $u = u(t,x)$ with initial data $u_0 = u_0(x)$ whose Sobolev norm $\|u_{0}\|_{s} = \varepsilon \lt \varepsilon_0$, during the time $|t| \leq \varepsilon^{-M}$, its Sobolev norm $\|u(t)\|_s$ remains bounded by $C_s \varepsilon$.

Article information

Taiwanese J. Math., Volume 21, Number 5 (2017), 1115-1132.

Received: 8 October 2016
Revised: 2 January 2017
Accepted: 8 January 2017
First available in Project Euclid: 1 August 2017

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Zentralblatt MATH identifier

Primary: 37K55: Perturbations, KAM for infinite-dimensional systems 35B10: Periodic solutions

KAM theory Hamiltonian systems Schrödinger equation Birkhoff normal form


Zhou, Shidi; Geng, Jiansheng. A Nekhoroshev Type Theorem of Higher Dimensional Nonlinear Schrödinger Equations. Taiwanese J. Math. 21 (2017), no. 5, 1115--1132. doi:10.11650/tjm/7951. https://projecteuclid.org/euclid.twjm/1501599185

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  • D. Bambusi, Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations, Math. Z. 230 (1999), no. 2, 345–387.
  • ––––, On long time stability in Hamiltonian perturbations of non-resonant linear PDEs, Nonlinearity 12 (1999), no. 4, 823–850.
  • ––––, An averaging theorem for quasilinear Hamiltonian PDEs, Ann. Henri Poincaré, 4 (2003), no. 4, 685–712.
  • ––––, Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys. 234 (2003), no. 2, 253–285.
  • D. Bambusi and B. Grébert, Birkhoff normal form for partial differential equations with tame modulus, Duke Math. J. 135 (2006), no. 3, 507–567.
  • G. Benettin, L. Galgani and A. Giorgilli, A proof of Nekhoroshev's theorem for the stability times in nearly integrable Hamiltonian systems, Celestial Mech. 37 (1985), no. 1, 1–25.
  • J. Bourgain, On diffusion in high-dimensional Hamiltonian systems and PDE, J. Anal. Math. 80 (2000), 1–35.
  • 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), no. 1, 39–113.
  • L. H. Eliasson, B. Grébert and S. B. Kuksin, KAM for the nonlinear beam equation, Geom. Funct. Anal. 26 (2016), no. 6, 1588–1715.
  • L. H. Eliasson and S. B. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. of Math. (2) 172 (2010), no. 1, 371–435.
  • E. Faou and B. Grébert, A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus, Anal. PDE 6 (2013), no. 6, 1243–1262.
  • J. Geng and J. You, A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Comm. Math. Phys. 262 (2006), no. 2, 343–372.
  • J. Geng, X. Xu and J. You, An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation, Adv. Math. 226 (2011), no. 6, 5361–5402.
  • A. Giorgilli and L. Galgani, Rigorous estimates for the series expansions of Hamiltonian perturbation theory, Celestial Mech. 37 (1985), no. 2, 95–112.
  • B. Grébert, Birkhoff normal form and Hamiltonian PDEs, in Partial Differential Equations and Applications, 1–46, Sémin. Congr. 15, Soc. Math. France, Paris, 2007.
  • M. Guardia, Growth of Sobolev norms in the cubic nonlinear Schrödinger equation with a convolution potential, Comm. Math. Phys. 329 (2014), no. 1, 405–434.
  • M. Guardia, E. Haus and M. Procesi, Growth of Sobolev norms for the analytic NLS on $\mb{T}^2$, Adv. Math. 301 (2016), 615–692.
  • M. Guardia and V. Kaloshin, Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 1, 71–149.
  • N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspehi Mat. Nauk 32 (1977), no. 6, 1–65.
  • J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z. 213 (1993), no. 1, 187–216.
  • M. Procesi and C. Procesi, A normal form for the Schrödinger equation with analytic non-linearities, Comm. Math. Phys. 312 (2012), no. 2, 501–557.
  • C. Procesi and M. Procesi, A KAM algorithm for the resonant non-linear Schrödinger equation, Adv. Math. 272 (2015), 399–470.
  • X. Yuan and J. Zhang, Long time stability of Hamiltonian partial differential equations, SIAM J. Math. Anal. 46 (2014), no. 5, 3176–3222.