Breathers and the dynamics of solutions to the KdV type equations

TL;DR

This paper characterizes conditions under which the generalized KdV equation lacks breather solutions and shows that solutions decay to zero over time in certain regions, regardless of supercritical scattering properties.

Contribution

It identifies a broad class of nonlinearities that prevent breather solutions and proves decay of bounded solutions in a specific dispersive region, independent of supercritical scattering.

Findings

No breathers or small breathers exist for certain nonlinearities.

Bounded solutions decay to zero in a region expanding as t^{1/2}.

Results hold regardless of supercritical scattering properties.

Abstract

In this paper our first aim is to identify a large class of non-linear functions $\,f(\cdot)\,$ 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 $L^1\cap H^1$ 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 $x\ll t$. 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.