# The effect of a positive bound state on the KdV solution. A case study

## TL;DR

This paper investigates how a positive bound state influences the solution of the KdV equation for a specific oscillatory potential, revealing a new bounded term in the solution related to the positive eigenvalue.

## Contribution

It demonstrates that the KdV equation admits a classical solution for a potential with a positive eigenvalue, introducing a novel bounded term akin to a positon, using Hankel operators.

## Key findings

- The solution includes a new bounded term due to the positive eigenvalue.
- The solution can be expressed in closed form using Hankel operators.
- The approach extends inverse scattering methods to non-rapidly decaying potentials.

## 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.