Well-Posedness of the Nonlinear Schrödinger Equations on the Line

We study the well-posedness of the Cauchy problem of the nonlinear Schrödinger equations on the line with two types of cubic nonlinearities with initial data in rough Sobolev spaces. First, using basic Fourier analysis, we solve the forced linear problem to obtain a solution formula in terms of the data and the forcing. Then, we estimate this solution in Bourgain spaces and bound it by the Sobolev norm of the data and different Bourgain norms of the forcing. This suggests that the iteration map obtained from the solution formula of the linear problem via replacing the forcing by the cubic nonlinearities can become a contraction on an appropriate solution space provided that we can prove the required trilinear estimates. We show that for nonlinearities $u|u|^2$ and $\bar{u}|u|^2$, the Cauchy problem of the nonlinear Schrödinger equation on the line is locally well-posed for sufficiently small initial data in $H_x^s(\mathbb{R})$ for $s > 0$.