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 Schrödinger 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 [H. Sunagawa: Osaka J. Math. 43 (2006), 771-789], in which the gauge-invariant nonlinearity was treated.