On the Cauchy problem for the super Korteweg–de Vries system

Abstract

Considered here is the super Korteweg–de Vries system: {∂tu+∂x3u+12∂x(u2)+12∂x2(v2)=0,∂tv+∂x3v+∂x(uv)=0. For small initial data, we show local well-posedness in Hs(R)∩H1(x2dx)×Hs(R)∩H1(x2dx), s≥3 integer, and for arbitrary large initial data we prove local well-posedness in Hs(R)∩H4(x2dx)×Hs(R)∩H4(x2dx), s≥8 integer. To obtain our main result we improve some linear estimates due to Kenig and Staffilani (1997) [5].