Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm

J. Colliander; M. Keel; G. Staffilani; H. Takaoka; T. Tao

doi:10.3934/cpaa.2003.2.33

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2003, Volume 2, Issue 1: 33-50. Doi: 10.3934/cpaa.2003.2.33 open access

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Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm

1.
University of Toronto, Toronto, Ontario, M5S 2E4
2.
University of Minnesota
3.
Massachusetts Institute of Technology
4.
Kobe University
5.
University of California, Los Angeles

Received: November 30, 2002

Published: October 2003

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Abstract

We continue the study (initiated in [18]) of the orbital stabilityof the ground state cylinder for focussing non-linear Schrödinger equationsin the $H^s(\R^n)$ norm for $1-\varepsilon < s < 1$, for small $\varepsilon$. In the $L^2$-subcritical case weobtain a polynomial bound for the time required to move away from theground state cylinder. If one is only in the $H^1$-subcritical casethen we cannot show this, but for defocussing equations we obtain global well-posedness andpolynomial growth of $H^s$ norms for $s$ sufficiently close to 1.

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Mathematics Subject Classification: 35Q53, 42B35, 37K10.

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