New direct linearizations for KdV and solutions of the other Cauchy problem

The author showed previously that there are several equations that “directly linearize” the Korteweg de Vries equation. They may give different classes of solutions and they correspond either to generalized Gelfand Levitan or generalized Marchenko inversion equations. The idea is developed here with a precise goal: solving by linear methods the “other Cauchy problem” for KdV, i.e., the boundary value problem where the solution is known at fixed x, together with its two first derivatives. After several new direct linearization equations are given and analyzed, the one that solves the problem is eventually obtained. It also corresponds to a new inverse spectral problem, whose scalar equations are fourth order, and that is first studied in the Gelfand Levitan form, and then studied and completely solved in the Marchenko form. The methods given here can be extended probably to most nonlinear integrable equations, and suggest several new problems.