January  2003, 9(1): 31-54. doi: 10.3934/dcds.2003.9.31

Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm

1. 

University of Toronto, Toronto, Ontario, M5S 2E4

2. 

University of Minnesota

3. 

Massachusetts Institute of Technology

4. 

Hokkaido University, Japan

5. 

University of California, Los Angeles

Received  May 2002 Revised  October 2002 Published  November 2002

We study the long-time behaviour of the focusing cubic NLS on $\mathbf R$ in the Sobolev norms $H^s$ for $0 < s < 1$. We obtain polynomial growth-type upper bounds on the $H^s$ norms, and also limit any orbital $H^s$ instability of the ground state to polynomial growth at worst; this is a partial analogue of the $H^1$ orbital stability result of Weinstein [27], [26]. In the sequel to this paper we generalize this result to other nonlinear Schrödinger equations. Our arguments are based on the "$I$-method" from earlier papers [9]-[15] which pushes down from the energy norm, as well as an "upside-down $I$-method" which pushes up from the $L^2$ norm.
Citation: J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao. Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 31-54. doi: 10.3934/dcds.2003.9.31
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