Spatial Analyticity of solutions to integrable systems. I. The KdV case (1109.6084v1)

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.