Well- and Ill-posedness of the Cauchy problem for derivative fractional nonlinear Schr\"odinger equations on the torus

TL;DR

This paper investigates the well- and ill-posedness of derivative fractional nonlinear Schrödinger equations on the torus, establishing conditions for well-posedness and demonstrating non-existence of solutions in certain cases.

Contribution

It extends the known well-posedness criteria to derivative fractional NLS without using gauge transformations, employing a modified energy method and correction terms.

Findings

Necessary and sufficient condition for well-posedness matches that of semi-linear Schrödinger equations.

Constructs correction terms for the modified energy when fractional order is between 1 and 1.5.

Proves non-existence of solutions using a Cauchy-Riemann-type operator in nonlinear interactions.

Abstract

We consider the Cauchy problem for derivative fractional Schr\"odinger equations (fNLS) on the torus $\mathbb T$. Recently, the second and third authors established a necessary and sufficient condition on the nonlinearity for well-posedness of semi-linear Schr\"odinger equations on $\mathbb T$. 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 Schr\"odinger 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 $\frac 32$. 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.