The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces H (R) defined by the norm
Local well-posedness for the th equation is shown in the parameter range 2 ≥ > 1, ≥ 2 -1 2 . The proof uses an appropriate variant of the Fourier restriction norm method. A counterexample is discussed to show that the Cauchy problem for equations of this type is in general ill-posed in the C 0 -uniform sense, if < 2 -1 2 . The results for = 2 -so far in the literature only if = 1 (mKdV) or = 2 -can be combined with the higher order conservation laws for the mKdV equation to obtain global well-posedness of the th equation in H (R) for ≥ +1 2 , if is odd, and for ≥ 2 , if is even. -The Cauchy problem for the th equation in the KdV hierarchy with data in H (R) cannot be solved by Picard iteration, if > 2 2 -1 , independent of the size of ∈ R. Especially for ≥ 2 we have C 2 -ill-posedness in H (R). With similar arguments as used before in the mKdV context it is shown that this problem is locally well-posed in
For KdV itself the lower bound on is pushed further down to > max (-1 2 -1 2 -1 4 -11 8 ), where ∈ (1 2). These results rely on the contraction mapping principle, and the flow map is real analytic.