Breathers and the dynamics of solutions to the KdV type equations

Abstract:
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^1 ∩ 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 x≪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.