CLASSICAL SOLUTIONS OF THE KORTEWEG-deVRIES EQUATION FOR NON-SMOOTH INITIAL DATA VIA INVERSE SCATTERING

ABSTRACT C The Cauchy problem for the Korteweg-deVries equation (KdV for short) q t(x,t) + q xx(x,t)- 6q(x,t)qx (x,t) = 0 q(x,0) =Q(x)-.is solved classically under the single assumption + (1 IxI) 41Q(x) dx <-for t> 0 via the so-called "inverse scattering method". This approach, originating with Gardner, Greene, Kruskal, and Miura (91, relates the KdV equation to the one-dimensional Schrdinger equation: (**)-f"(x,k) + u(x)f(x,k) = k2f(x,k).!By considering the effect on the scattering data associated to the Schrodinger equation (**) when the potential u(x) evolves in t according to the KdV equation (*), one obtains a linear evolution equation for the scattering data. The inverse scattering method of solving (*) consists of calculating the scattering data for the initial value Q(x), letting it evolve to time t, and then recovering q(x,t) from the evolved scattering data. Recently, P. Deift and E. Trubowitz [7] presented a new method for solving the inverse scattering problem (obtaining the potential from it