Well-posedness for one-dimensional derivative nonlinear Schr\"odinger equations

In this paper, we investigate the one-dimensional derivative nonlinear Schr"odinger equations of the form $i{u}_t-u_{xx}+i\lambda \text{abs}{u}^k{u}_x=0$ with non-zero $\lambda \in \text{Real}$ and any real number $k\text{gs}5$. We establish the local well-posedness of the Cauchy problem with any initial data in $H^{1\mathrm{/}2}$ by using the gauge transformation and the Littlewood-Paley decomposition.