Classical solutions of the korteweg—devries equation for non—smooth initial data via inverse scattering*

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Abstract
The Cauchy problem for the Korteweg—deVries equation (KdV for short) (*) is solved classically for t>0 under the single assumption for via the so—called “inverse scattering method”. This approach, originating with Gardner, Greene Kruskal, and Miura [11], relates the KdV equation to the one—dimensional Schrödinger equation: (**) By considering the effect on the scattering data associated to the Schrödinger equation (**) when the potential u(x) evolves in t according to the KfV equation (*), one obtains a linear evolution equation for the scatering 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) form the evolved scattering data. Recently, P. Deift and E. Trubowitz [9] presented a new method for solving the inverse scattering problem (obtaining the potential form its scattering data). Our solution of the KdV initial value problem uses this approach to construct a classical solution under the assumption stated above.

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