Local well-posedness for the super Korteweg-de Vries equation

We prove local well-posedness in weighted Sobolev spaces: Xs,3≔(Hs(R)∩H3(x2dx))×(Hs(R)∩H3(x2dx)),s≥5 integer, provided that the initial data is small enough, and also in Xs,11≔(Hs(R)∩H11(x2dx))×(Hs(R)∩H11(x2dx)) with s≥15 integer, for arbitrary initial data. The main ingredients of the proof for the first case are new estimates describing the smoothing effect of Kato type for the KdV group {W(t)}t∈R; that for the second case is via a change of variable performed and a deduction of new smoothing effects related to the KdV [C. Kenig, G. Staffilani, Local well-posedness for higher order nonlinear dispersive systems, J. Fourier Anal. Appl. 3 (4) (1997)].