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arXiv · 1803.05475

Breathers and the dynamics of solutions to the KdV type equations

Claudio Muñoz·Gustavo Ponce
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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.

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Claudio Muñoz, Gustavo Ponce. 2018-03-14. Breathers and the dynamics of solutions to the KdV type equations. https://doi.org/10.1007/s00220-018-3206-9

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