# American Institute of Mathematical Sciences

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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