We show that the Cauchy problem for the KdV equation can be solved by the inverse scattering transform (IST) for any initial data bounded from below, decaying sufficiently rapidly at +∞ but unrestricted otherwise. Thus our approach does not require any boundary condition at −∞.
Rybkin, A. (2018). KdV equation beyond standard assumptions on initial data. Physica D: Nonlinear Phenomena, 365, 1–11. https://doi.org/10.1016/j.physd.2017.10.005