Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity

Abstract

We show the well-posedness in $H^{\frac 12 }$ of the Cauchy problem for a certain class of one dimensional nonlinear Schrödinger equations with the derivative nonlinearity. This is an improvement of results in $H^1$ by N. Hayashi and T. Ozawa [2,3,4,20]. Our results can cover the derivative nonlinear Schrödinger equation. Our proof is based on the Fourier restriction norm method and the gauge transformation.