### Abstract

We study the long-time behaviour of the focusing cubic NLS on 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.

Original language | English (US) |
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Pages (from-to) | 31-54 |

Number of pages | 24 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 9 |

Issue number | 1 |

State | Published - Jan 1 2003 |

### Keywords

- Orbital stability
- Schrödinger equation
- Upper bound on sobolev norms

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## Cite this

Colliander, J., Keel, M., Staffilani, G., Takaoka, H., & Tao, T. (2003). Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm.

*Discrete and Continuous Dynamical Systems*,*9*(1), 31-54.