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

Jim Colliander·Mark Keel·Gigliola Staffilani·Hideo Takaoka·Terence Tao
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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. In the sequel to this paper we generalize this result to other nonlinear Schrödinger equations. Our arguments are based on the ``$I$-method'' from our earlier papers, which pushes down from the energy norm, as well as an ``upside-down $I$-method'' which pushes up from the $L^2$ norm.

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Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka, Terence Tao. 2002-11-12. Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm. https://arxiv.org/abs/math/0206218

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