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 & 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,, \emph{Comm. Math. Phys.}, 234 (2003), 253. doi: 10.1007/s00220-002-0774-4. Google Scholar

[2]

D. Bambusi and B. Grebert, Birkhoff normal form for PDE's with tame modulus,, \emph{Duke Math. J.}, 135 (2006), 507. doi: 10.1215/S0012-7094-06-13534-2. Google Scholar

[3]

J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE,, \emph{Intern. Math. Res. Notices}, 6 (1996), 277. doi: 10.1155/S1073792896000207. Google Scholar

[4]

J. Bourgain, On diffusion in High-dimensional Hamiltonian systems and PDE,, \emph{J. d'analyse Math.}, 80 (2000), 1. doi: 10.1007/BF02791532. Google Scholar

[5]

J. Bourgain, Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations,, \emph{Ergodic Theory Dynam. Syst.}, 24 (2004), 1331. doi: 10.1017/S0143385703000750. Google Scholar

[6]

D. Cohen, E. Hairer and C. Lubich, Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions,, \emph{Arch. Rational Mech. Anal.}, 187 (2008), 341. doi: 10.1007/s00205-007-0095-z. Google Scholar

[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,, \emph{J. d'analyse Math.}, 118 (2012), 55. doi: 10.1007/s11854-012-0029-z. Google Scholar

[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,, \emph{Invent. Math.}, 181 (2010), 39. doi: 10.1007/s00222-010-0242-2. Google Scholar

[9]

L. H. Eliasson and S. B. Kuksin, KAM for nonlinear Schrödinger equation,, \emph{Annals of Math.}, 172 (2010), 371. doi: 10.4007/annals.2010.172.371. Google Scholar

[10]

E. Faou, and B. Grebert, A Nekhoroshev type theorem for the nonlinear Schrödinger equation on the d-dimensional torus,, \emph{Analysis and PDE}, 6 (2013), 1243. doi: 10.2140/apde.2013.6.1243. Google Scholar

[11]

B. Grebert, Birkhoff normal form and Hamiltonian PDEs,, \emph{Partial Differential Equations and Applications}, (2007), 1. Google Scholar

[12]

B. Grebert, T. Kappeler and J. Pöschel, Normal form theory for the NLS equatons,, preprint, (). Google Scholar

[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}., (). Google Scholar

[14]

M. Guardia, Growth of Sobolev norms in the cubic nonlinear Schrödinger equation with a convolution potential,, \emph{Comm. Math. Phys.}, 329 (2014), 405. doi: 10.1007/s00220-014-1977-1. Google Scholar

[15]

M. Guardia, and V. Kaloshin, Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation,, \emph{J. Eur. Math. Soc.}, 17 (2015), 71. doi: 10.4171/JEMS/499. Google Scholar

[16]

Z. Hani, B. Pausader, N. Tzvetkov and N. Visciglia, Modified scattering for the cubic Schrödinger equation on product spaces and applications,, \emph{Forum of mathematics, 3 (2015). doi: 10.1017/fmp.2015.5. Google Scholar

[17]

T. Kappeler and J. Pöschel, KdV and KAM,, A Series of Modern Surveys in Mathematics, (2003). doi: 10.1007/978-3-662-08054-2. Google Scholar

[18]

S. B. Kuksin, Oscillations in space-periodic nonlinear Schrodinger equations,, \emph{Geom. Funct. Anal.}, 7 (1997), 338. doi: 10.1007/PL00001622. Google Scholar

[19]

V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $R$,, \emph{Indiana Univ. Math. J.}, 60 (2011), 1487. doi: 10.1512/iumj.2011.60.4399. Google Scholar

[20]

V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations,, \emph{Discret. Contin. Dyn. Syst.}, 32 (2012), 3733. doi: 10.3934/dcds.2012.32.3733. Google Scholar

[21]

G. Staffilani, On the growth of high sobolev norms of solutions for kdv and Schrödinger equations,, \emph{Duke Math. J.}, 86 (1997), 109. doi: 10.1215/S0012-7094-97-08604-X. Google Scholar

[22]

W.-M. Wang, Long time anderson localization for the nonlinear random Schrödinger equation,, \emph{J. Stat. Physics}, 134 (2009), 953. doi: 10.1007/s10955-008-9649-1. Google Scholar

show all references

References:
[1]

D. Bambusi, Birkhoff normal form for some nonlinear PDEs,, \emph{Comm. Math. Phys.}, 234 (2003), 253. doi: 10.1007/s00220-002-0774-4. Google Scholar

[2]

D. Bambusi and B. Grebert, Birkhoff normal form for PDE's with tame modulus,, \emph{Duke Math. J.}, 135 (2006), 507. doi: 10.1215/S0012-7094-06-13534-2. Google Scholar

[3]

J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE,, \emph{Intern. Math. Res. Notices}, 6 (1996), 277. doi: 10.1155/S1073792896000207. Google Scholar

[4]

J. Bourgain, On diffusion in High-dimensional Hamiltonian systems and PDE,, \emph{J. d'analyse Math.}, 80 (2000), 1. doi: 10.1007/BF02791532. Google Scholar

[5]

J. Bourgain, Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations,, \emph{Ergodic Theory Dynam. Syst.}, 24 (2004), 1331. doi: 10.1017/S0143385703000750. Google Scholar

[6]

D. Cohen, E. Hairer and C. Lubich, Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions,, \emph{Arch. Rational Mech. Anal.}, 187 (2008), 341. doi: 10.1007/s00205-007-0095-z. Google Scholar

[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,, \emph{J. d'analyse Math.}, 118 (2012), 55. doi: 10.1007/s11854-012-0029-z. Google Scholar

[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,, \emph{Invent. Math.}, 181 (2010), 39. doi: 10.1007/s00222-010-0242-2. Google Scholar

[9]

L. H. Eliasson and S. B. Kuksin, KAM for nonlinear Schrödinger equation,, \emph{Annals of Math.}, 172 (2010), 371. doi: 10.4007/annals.2010.172.371. Google Scholar

[10]

E. Faou, and B. Grebert, A Nekhoroshev type theorem for the nonlinear Schrödinger equation on the d-dimensional torus,, \emph{Analysis and PDE}, 6 (2013), 1243. doi: 10.2140/apde.2013.6.1243. Google Scholar

[11]

B. Grebert, Birkhoff normal form and Hamiltonian PDEs,, \emph{Partial Differential Equations and Applications}, (2007), 1. Google Scholar

[12]

B. Grebert, T. Kappeler and J. Pöschel, Normal form theory for the NLS equatons,, preprint, (). Google Scholar

[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}., (). Google Scholar

[14]

M. Guardia, Growth of Sobolev norms in the cubic nonlinear Schrödinger equation with a convolution potential,, \emph{Comm. Math. Phys.}, 329 (2014), 405. doi: 10.1007/s00220-014-1977-1. Google Scholar

[15]

M. Guardia, and V. Kaloshin, Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation,, \emph{J. Eur. Math. Soc.}, 17 (2015), 71. doi: 10.4171/JEMS/499. Google Scholar

[16]

Z. Hani, B. Pausader, N. Tzvetkov and N. Visciglia, Modified scattering for the cubic Schrödinger equation on product spaces and applications,, \emph{Forum of mathematics, 3 (2015). doi: 10.1017/fmp.2015.5. Google Scholar

[17]

T. Kappeler and J. Pöschel, KdV and KAM,, A Series of Modern Surveys in Mathematics, (2003). doi: 10.1007/978-3-662-08054-2. Google Scholar

[18]

S. B. Kuksin, Oscillations in space-periodic nonlinear Schrodinger equations,, \emph{Geom. Funct. Anal.}, 7 (1997), 338. doi: 10.1007/PL00001622. Google Scholar

[19]

V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $R$,, \emph{Indiana Univ. Math. J.}, 60 (2011), 1487. doi: 10.1512/iumj.2011.60.4399. Google Scholar

[20]

V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations,, \emph{Discret. Contin. Dyn. Syst.}, 32 (2012), 3733. doi: 10.3934/dcds.2012.32.3733. Google Scholar

[21]

G. Staffilani, On the growth of high sobolev norms of solutions for kdv and Schrödinger equations,, \emph{Duke Math. J.}, 86 (1997), 109. doi: 10.1215/S0012-7094-97-08604-X. Google Scholar

[22]

W.-M. Wang, Long time anderson localization for the nonlinear random Schrödinger equation,, \emph{J. Stat. Physics}, 134 (2009), 953. doi: 10.1007/s10955-008-9649-1. Google Scholar

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