Focusing Singularity in a Derivative Nonlinear Schr\"odinger Equation

We present a numerical study of a derivative nonlinear Schr"odinger equation with a general power nonlinearity, $\mathrm{\mid }\psi {\mathrm{\mid }}^{2\sigma }{\psi }_x$. In the $L^2$-supercritical regime, $\sigma >1$, our simulations indicate that there is a finite time singularity. We obtain a precise description of the local structure of the solution in terms of blowup rate and asymptotic profile, in a form similar to that of the nonlinear Schr"odinger equation with supercritical power law nonlinearity.