Meromorphic solutions to the KdV equation with non-decaying initial data supported on a left half line

Abstract

We show that under the Korteweg–de Vries flow an initial profile supported on (−∞, 0) from a very broad function class (without any decay assumption) instantaneously evolves into a meromorphic function with no poles on the real line. Our treatment is based on a suitable modification of the inverse scattering transform and a detailed investigation of the Titchmarsh–Weyl m-function. As a by-product, we improve some related results of others.