Local Well-posedness Of 1-D Nonlinear Second Order Schr?dinger Equation

The global and local well-posedness of the 1-dimensional second-order nonlinear Schr?dinger equation has a hot research topic in recent years.Dividing the frequency space by different frequency decomposition methods,in order to discuss the local well-posedness of equation.Various kinds of spaces are introduced,such as Bourgain space and Modulation space.This article discuss local well-posedness problem of this equation.Through the frequency uniform decomposion of a solution in the whole space,the global well-posedness estimate of the solution is converted into the unit nonlinear estimate.By discussing the relationship between different frequency and using the Strichartz estimates and the Bilinear Strichartz estimates,we obtained the local well-posedness of equation.