| Abstract: |
| In this paper our first aim is to identify a large class of non-linear functions $f$ for which the IVP for the generalized Korteweg-de Vries equation does not have breathers or ``small`` breathers solutions. Also we prove that all small, uniformly in time $L^1cap H^1$ bounded solutions to KdV and related perturbations must converge to zero, as time goes to infinity, locally in an increasing-in-time region of space of order $t^{1/2}$ around any compact set in space. This set is included in the linearly dominated dispersive region $xll t$. Moreover, we prove this result independently of the well-known supercritical character of KdV scattering. In particular, no standing breather-like nor solitary wave structures exists in this particular regime. |
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