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

TL;DR

This paper investigates the spatial analyticity of solutions to the KdV equation with nonsmooth initial data, revealing how the regularity depends on the decay rate of initial profiles using inverse scattering.

Contribution

It establishes the relationship between initial data decay rates and the resulting spatial analyticity of KdV solutions, including meromorphic and Gevrey regularity results.

Findings

Solutions become meromorphic if decay is fast enough (>1/2)

Solutions are meromorphic on a strip for =1/2

Solutions are Gevrey regular if <1/2

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.