# American Institute of Mathematical Sciences

March  2016, 15(2): 341-365. doi: 10.3934/cpaa.2016.15.341

## The stability of nonlinear Schrödinger equations with a potential in high Sobolev norms revisited

 1 Department of Applied Mathematics, Hankyong National University, Ansong 456-749 2 Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 335 Gwahangno, Yuseong-gu, Daejeon, Korea 305-701

Received  February 2015 Revised  November 2015 Published  January 2016

We consider the nonlinear Schrödinger equations with a potential on $\mathbb T^d$. For almost all potentials, we show the almost global stability in very high Sobolev norms. We apply an iteration of the Birkhoff normal form, as in the formulation introduced by Bourgain [4]. This result reproves a dynamical consequence of the infinite dimensional Birkhoff normal form theorem by Bambusi and Grebert [2].
Citation: Myeongju Chae, Soonsik Kwon. The stability of nonlinear Schrödinger equations with a potential in high Sobolev norms revisited. Communications on Pure and Applied Analysis, 2016, 15 (2) : 341-365. doi: 10.3934/cpaa.2016.15.341
##### References:
 [1] D. Bambusi, Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys., 234 (2003), 253-285. doi: 10.1007/s00220-002-0774-4. [2] D. Bambusi and B. Grebert, Birkhoff normal form for PDE's with tame modulus, Duke Math. J., 135 (2006), 507-567. doi: 10.1215/S0012-7094-06-13534-2. [3] J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Intern. Math. Res. Notices, 6 (1996), 277-304. doi: 10.1155/S1073792896000207. [4] J. Bourgain, On diffusion in High-dimensional Hamiltonian systems and PDE, J. d'analyse Math., 80 (2000), 1-34. doi: 10.1007/BF02791532. [5] J. Bourgain, Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations, Ergodic Theory Dynam. Syst., 24 (2004), 1331-1357. doi: 10.1017/S0143385703000750. [6] D. Cohen, E. Hairer and C. Lubich, Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions, Arch. Rational Mech. Anal., 187 (2008), 341-368. doi: 10.1007/s00205-007-0095-z. [7] J. Colliander, S. Kwon and T. Oh, A remark on normal forms and the "upside-down" I -method for periodic NLS: growth of higher Sobolev norms, J. d'analyse Math., 118 (2012), 55-82. doi: 10.1007/s11854-012-0029-z. [8] 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. doi: 10.1007/s00222-010-0242-2. [9] L. H. Eliasson and S. B. Kuksin, KAM for nonlinear Schrödinger equation, Annals of Math., 172 (2010), 371-435. doi: 10.4007/annals.2010.172.371. [10] E. Faou, and B. Grebert, A Nekhoroshev type theorem for the nonlinear Schrödinger equation on the d-dimensional torus, Analysis and PDE, 6 (2013), 1243-1262. doi: 10.2140/apde.2013.6.1243. [11] B. Grebert, Birkhoff normal form and Hamiltonian PDEs, Partial Differential Equations and Applications, 1-46, Sémin. Congr., 15, Soc. Math. France, Paris, 2007. [12] B. Grebert, T. Kappeler and J. Pöschel, Normal form theory for the NLS equatons,, preprint, (). [13] B. Grebert, E. Paturel and L. Thomann, Modiifed scattering for the cubic Schrödinger equation on product spaces: the non resonance case, preprint,, \arXiv{1502.07699}., (). [14] M. Guardia, Growth of Sobolev norms in the cubic nonlinear Schrödinger equation with a convolution potential, Comm. Math. Phys., 329 (2014), 405-434. doi: 10.1007/s00220-014-1977-1. [15] M. Guardia, and V. Kaloshin, Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation, J. Eur. Math. Soc., 17 (2015), 71-149. doi: 10.4171/JEMS/499. [16] Z. Hani, B. Pausader, N. Tzvetkov and N. Visciglia, Modified scattering for the cubic Schrödinger equation on product spaces and applications, Forum of mathematics, Pi, 3 (2015), e4, 63pages. doi: 10.1017/fmp.2015.5. [17] T. Kappeler and J. Pöschel, KdV and KAM, A Series of Modern Surveys in Mathematics, Springer, 2003. doi: 10.1007/978-3-662-08054-2. [18] S. B. Kuksin, Oscillations in space-periodic nonlinear Schrodinger equations, Geom. Funct. Anal., 7 (1997), 338-363. doi: 10.1007/PL00001622. [19] V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $R$, Indiana Univ. Math. J., 60 (2011), 1487-1516. doi: 10.1512/iumj.2011.60.4399. [20] V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations, Discret. Contin. Dyn. Syst., 32 (2012), 3733-3771. doi: 10.3934/dcds.2012.32.3733. [21] G. Staffilani, On the growth of high sobolev norms of solutions for kdv and Schrödinger equations, Duke Math. J., 86 (1997), 109-142. doi: 10.1215/S0012-7094-97-08604-X. [22] W.-M. Wang, Long time anderson localization for the nonlinear random Schrödinger equation, J. Stat. Physics, 134 (2009), 953-968. doi: 10.1007/s10955-008-9649-1.

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##### References:
 [1] D. Bambusi, Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys., 234 (2003), 253-285. doi: 10.1007/s00220-002-0774-4. [2] D. Bambusi and B. Grebert, Birkhoff normal form for PDE's with tame modulus, Duke Math. J., 135 (2006), 507-567. doi: 10.1215/S0012-7094-06-13534-2. [3] J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Intern. Math. Res. Notices, 6 (1996), 277-304. doi: 10.1155/S1073792896000207. [4] J. Bourgain, On diffusion in High-dimensional Hamiltonian systems and PDE, J. d'analyse Math., 80 (2000), 1-34. doi: 10.1007/BF02791532. [5] J. Bourgain, Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations, Ergodic Theory Dynam. Syst., 24 (2004), 1331-1357. doi: 10.1017/S0143385703000750. [6] D. Cohen, E. Hairer and C. Lubich, Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions, Arch. Rational Mech. Anal., 187 (2008), 341-368. doi: 10.1007/s00205-007-0095-z. [7] J. Colliander, S. Kwon and T. Oh, A remark on normal forms and the "upside-down" I -method for periodic NLS: growth of higher Sobolev norms, J. d'analyse Math., 118 (2012), 55-82. doi: 10.1007/s11854-012-0029-z. [8] 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. doi: 10.1007/s00222-010-0242-2. [9] L. H. Eliasson and S. B. Kuksin, KAM for nonlinear Schrödinger equation, Annals of Math., 172 (2010), 371-435. doi: 10.4007/annals.2010.172.371. [10] E. Faou, and B. Grebert, A Nekhoroshev type theorem for the nonlinear Schrödinger equation on the d-dimensional torus, Analysis and PDE, 6 (2013), 1243-1262. doi: 10.2140/apde.2013.6.1243. [11] B. Grebert, Birkhoff normal form and Hamiltonian PDEs, Partial Differential Equations and Applications, 1-46, Sémin. Congr., 15, Soc. Math. France, Paris, 2007. [12] B. Grebert, T. Kappeler and J. Pöschel, Normal form theory for the NLS equatons,, preprint, (). [13] B. Grebert, E. Paturel and L. Thomann, Modiifed scattering for the cubic Schrödinger equation on product spaces: the non resonance case, preprint,, \arXiv{1502.07699}., (). [14] M. Guardia, Growth of Sobolev norms in the cubic nonlinear Schrödinger equation with a convolution potential, Comm. Math. Phys., 329 (2014), 405-434. doi: 10.1007/s00220-014-1977-1. [15] M. Guardia, and V. Kaloshin, Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation, J. Eur. Math. Soc., 17 (2015), 71-149. doi: 10.4171/JEMS/499. [16] Z. Hani, B. Pausader, N. Tzvetkov and N. Visciglia, Modified scattering for the cubic Schrödinger equation on product spaces and applications, Forum of mathematics, Pi, 3 (2015), e4, 63pages. doi: 10.1017/fmp.2015.5. [17] T. Kappeler and J. Pöschel, KdV and KAM, A Series of Modern Surveys in Mathematics, Springer, 2003. doi: 10.1007/978-3-662-08054-2. [18] S. B. Kuksin, Oscillations in space-periodic nonlinear Schrodinger equations, Geom. Funct. Anal., 7 (1997), 338-363. doi: 10.1007/PL00001622. [19] V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $R$, Indiana Univ. Math. J., 60 (2011), 1487-1516. doi: 10.1512/iumj.2011.60.4399. [20] V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations, Discret. Contin. Dyn. Syst., 32 (2012), 3733-3771. doi: 10.3934/dcds.2012.32.3733. [21] G. Staffilani, On the growth of high sobolev norms of solutions for kdv and Schrödinger equations, Duke Math. J., 86 (1997), 109-142. doi: 10.1215/S0012-7094-97-08604-X. [22] W.-M. Wang, Long time anderson localization for the nonlinear random Schrödinger equation, J. Stat. Physics, 134 (2009), 953-968. doi: 10.1007/s10955-008-9649-1.
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