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(4)
nonlinear schrodinger equation
Normal Form
schrodinger equation
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A Remark on Normal Forms and the "Upsidedown" Imethod for periodic NLS: Growth of Higher Sobolev Norms
A Remark on Normal Forms and the "Upsidedown" Imethod for periodic NLS: Growth of Higher Sobolev Norms,James Colliander,Soonsik Kwon,Tadahiro Oh
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A Remark on Normal Forms and the "Upsidedown" Imethod for periodic NLS: Growth of Higher Sobolev Norms
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James Colliander
,
Soonsik Kwon
,
Tadahiro Oh
We study growth of higher Sobolev norms of solutions to the onedimensional periodic
nonlinear Schrodinger equation
(NLS). By a combination of the
normal form
reduction and the upsidedown Imethod, we establish \u(t)\_{H^s} \lesssim (1+t)^{\alpha (s1)+} with \alpha = 1 for a general power nonlinearity. In the quintic case, we obtain the above estimate with \alpha = 1/2 via the spacetime estimate due to Bourgain [4], [5]. In the cubic case, we concretely compute the terms arising in the first few steps of the
normal form
reduction and prove the above estimate with \alpha = 4/9. These results improve the previously known results (except for the quintic case.) In Appendix, we also show how Bourgain's idea in [4] on the
normal form
reduction for the quintic nonlinearity can be applied to other powers.
Published in 2010.
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References
(12)
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations
(
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)
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Journal:
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Remarks on stability and diffusion in highdimensional Hamiltonian systems and partial differential equations
(
Citations: 11
)
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Journal:
Ergodic Theory and Dynamical Systems  ERGOD THEOR DYN SYST
, vol. 24, no. 5, pp. 13311357, 2004
A remark on normal forms and the “ I method” for periodic NLS
(
Citations: 15
)
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Journal:
Journal D Analyse Mathematique  J ANAL MATH
, vol. 94, no. 1, pp. 125157, 2004
Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm
(
Citations: 7
)
Jim Colliander
,
Mark Keel
,
Gigliola Staffilani
,
Hideo Takaoka
,
Terence Tao
Published in 2002.
QuasiLinear Dynamics in Nonlinear Schrödinger Equation with Periodic Boundary Conditions
(
Citations: 5
)
M. Burak Erdoğan
,
Vadim Zharnitsky
Journal:
Communications in Mathematical Physics  COMMUN MATH PHYS
, vol. 281, no. 3, pp. 655673, 2008