Local well-posedness for the super Korteweg–de Vries equation Original Research Article

We establish some qualitative properties for the Cauchy problem associated with the super Korteweg–de Vries equation (super-KdV ) equation(super-KdV ) View the MathML source{∂tu+∂x3u+12∂xu2+12∂x2v2=0∂tv+∂x3v+∂x(uv)=0,x,t∈R. Turn MathJax on We prove local well-posedness in weighted Sobolev spaces: Xs,3≔(Hs(R)∩H3(x2dx))×(Hs(R)∩H3(x2dx)),Xs,3≔(Hs(R)∩H3(x2dx))×(Hs(R)∩H3(x2dx)), Turn MathJax on s≥5s≥5 integer, provided that the initial data is small enough, and also in Xs,11≔(Hs(R)∩H11(x2dx))×(Hs(R)∩H11(x2dx))Xs,11≔(Hs(R)∩H11(x2dx))×(Hs(R)∩H11(x2dx)) Turn MathJax on with s≥15s≥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{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)].