Well- and Ill-posedness of the Cauchy problem for derivative fractional nonlinear Schrödinger equations on the torus

We consider the Cauchy problem for derivative fractional Schrödinger equations (fNLS) on the torus T. Recently, the second and third authors established a necessary and sufficient condition on the nonlinearity for well-posedness of semi-linear Schrödinger equations on T. In this paper, we extend this result to derivative fNLS. More precisely, we prove that the necessary and sufficient condition on the nonlinearity is the same as that for semi-linear Schrödinger equations. However, since we can not employ a gauge transformation for derivative fNLS, we use the modified energy method to prove well-posedness. We need to inductively construct correction terms for the modified energy when the fractional Laplacian is of order between $1$ and 32. For the ill-posedness, we prove the non-existence of solutions to the Cauchy problem by exploiting a Cauchy-Riemann-type operator that appears in nonlinear interactions.