Formal initial value problem of the Korteweg-de Vries equation

We study the initial value problem of the Korteweg-de Vries (KdV) equation on a space of generalized formal power series. We derive an explicit expression of the solution of the KdV equation with an arbitrary initial condition, using a recursively defined sequence of rational functions. From this result one can explain the formal analogues of the direct and inverse scattering transforms, relating the given initial condition to the solution of the formal Gelfand-Levitan-Marchenko equation.