The asymptotic behavior of solutions of the korteweg-deVries equation evolving from very irregular data

Abstract

Let u(x, t) be a solution of KdV constructed by the inverse scattering method with u(x, 0) = μ(x), a measure. Under general conditions on μ, there is a pure n-soliton solution of KdV, us(x, t), such that u(x, t) − us(x, t) = O(t−1/3) as t → ∞ uniformly in any [x0, + ∞). If n = 0, then the convergence rate is O(t−2/3) rather than O(t−1/3).