|XXXV Workshop on Geometric Methods in Physics
The Cauchy problem for the KdV equation with no decay at minus infinity
Using techniques of Hankel operators we prove that the KdV equation is well-posed for initial data which have nearly unrestricted behavior at minus infinity and short range decay at plus infinity. We also develop the inverse scattering transform for such initial profiles in term of the Hankel operator a Hankel operator which symbol is conveniently represented in terms of the scattering data for the Schrodinger operator in the Lax pair. The spectral properties of this Schrodinger operator can be directly translated into the spectral properties of the Hankel operator. The latter then yield properties of the solutions to the KdV equation through explicit formulas. This allows us to recover and improve on many already known results as well as a variety of new ones. We can effortlessly include into our picture low regularity that is far beyond what standard harmonic analytic techniques can handle. Main results discussed in the talk are optimal.