Local Existence Theory for Derivative Nonlinear Schr\"{o}dinger Equations with Non-Integer Power Nonlinearities

We study a derivative nonlinear Schr"{o}dinger equation, allowing non-integer powers in the nonlinearity, $\mathrm{\mid }u{\mathrm{\mid }}^{2\sigma }{u}_x$. Making careful use of the energy method, we are able to establish short-time existence of solutions with initial data in the energy space, $H^1$. For more regular initial data, we establish not just existence of solutions, but also well-posedness of the initial value problem. These results hold for real-valued $\sigma \ge 1,$ while prior existence results in the literature require integer-valued $\sigma$ or $\sigma$ sufficiently large ($\sigma \ge 5\mathrm{/}2$), or use higher-regularity function spaces.