Local Existence Theory for Derivative Nonlinear Schrödinger Equations with Non-Integer Power Nonlinearities

Abstract

We study a derivative nonlinear Schrödinger equation, allowing non-integer powers in the nonlinearity, $|u|^{2σ} u_x$. 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 $σ\geq 1,$ while prior existence results in the literature require integer-valued $σ$ or $σ$ sufficiently large ($σ\geq 5/2$), or use higher-regularity function spaces.