KdV equation beyond standard assumptions on initial data

TL;DR

This paper extends the applicability of the inverse scattering transform to solve the KdV equation for a broader class of initial data, specifically those bounded from below and rapidly decaying at plus infinity, without boundary conditions at minus infinity.

Contribution

It demonstrates that the IST method can be used for initial data with minimal restrictions, beyond the standard assumptions, by removing the need for boundary conditions at minus infinity.

Findings

Successfully solves KdV with less restrictive initial data

Extends IST applicability to broader initial data classes

No boundary condition required at minus infinity

Abstract

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 plus infinity, but unrestricted otherwise. Thus our approach doesn't require any boundary condition at minus infinity.