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Nevertheless, via symplectic considerations, we show that solutions remain $O(\\epsilon \\la t\\ra^{1/2})$ close to a soliton on an $O(\\epsilon^{-1})$ time scale. Furthermore, we show that the soliton parameters can be chosen to evolve according to specific exact ODEs on the shorter, but still dynamically relevant, time scale $O(\\epsilon^{-1/2})$. 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This type of perturbation is non-Hamiltonian. Nevertheless, via symplectic considerations, we show that solutions remain $O(\\epsilon \\la t\\ra^{1/2})$ close to a soliton on an $O(\\epsilon^{-1})$ time scale. Furthermore, we show that the soliton parameters can be chosen to evolve according to specific exact ODEs on the shorter, but still dynamically relevant, time scale $O(\\epsilon^{-1/2})$. 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