Breathers and the Dynamics of Solutions in KdV Type Equations

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 uniformly in time L-1 H-1 bounded solutions to KdV and related small 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.