The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension

Abstract: Consider the initial value problem for cubic derivative nonlinear Schr\"odinger equations in one space dimension. We provide a detailed lower bound estimate for the lifespan of the solution, which can be computed explicitly from the initial data and the nonlinear term. This is an extension and a refinement of the previous work by one of the authors where the gauge-invariant nonlinearity was treated.