Spatial Analyticity of solutions to integrable systems. I. The KdV case

Abstract

We are concerned with the Cauchy problem for the KdV equation for nonsmooth locally integrable initial profiles q's which are, in a certain sense, essentially bounded from below and q(x)=O(e^{-cx^{{\epsilon}}}),x\rightarrow+\infty, with some positive c and {\epsilon}. Using the inverse scattering transform, we show that the KdV flow turns such initial data into a function which is (1) meromorphic (in the space variable) on the whole complex plane if {\epsilon}>1/2, (2) meromorphic on a strip around the real line if {\epsilon}=1/2, and (3) Gevrey regular if {\epsilon}<1/2. Note that q's need not have any decay or pattern of behavior at -\infty.