Local Existence Theory for Derivative Nonlinear Schrödinger Equations with Noninteger Power Nonlinearities

We study a derivative nonlinear Schrödinger equation, allowing noninteger powers in the nonlinearity, $|u|^{2\sigma} u_x$. Our main theorem is short-time existence of solutions with initial data in the energy space, $H^1;$ this is achieved by a careful use of the energy method. For more regular initial data, we establish not just the existence of solutions but also the well-posedness of the initial value problem. These results hold for real-valued $\sigma\geq 1,$ while prior existence results in the literature require integer-valued $\sigma$ or $\sigma$ sufficiently large ($\sigma \geq 5/2$), use higher-regularity function spaces, or impose a smallness condition on the initial data.