The effect of a positive bound state on the KdV solution. A case study
Abstract
We consider a slowly decaying oscillatory potential such that the corresponding 1D Schr\"odinger operator has a positive eigenvalue embedded into the absolutely continuous spectrum. This potential does not fall into a known class of initial data for which the Cauchy problem for the Korteweg-de Vries (KdV) equation can be solved by the inverse scattering transform. We nevertheless show that the KdV equation with our potential does admit a closed form classical solution in terms of Hankel operators. Comparing with rapidly decaying initial data our solution gains a new term responsible for the positive eigenvalue. To some extend this term resembles a positon (singular) solution but remains bounded. Our approach is based upon certain limiting arguments and techniques of Hankel operators.
Cite
@article{arxiv.1905.08373,
title = {The effect of a positive bound state on the KdV solution. A case study},
author = {Alexei Rybkin},
journal= {arXiv preprint arXiv:1905.08373},
year = {2019}
}