Solutions to the Cubic Schrödinger Equation by the Inverse Scattering Method

Weak solutions to the cubic Schrödinger equation are constructed by the inverse scattering method for a large class of initial data $u_0 $ such that $(1 + |x|^\alpha )u_0 \in L^1 (\mathbb{R}) \cap L^2 (\mathbb{R})$ and $u_0 \in H^\alpha (\mathbb{R})$ for an $\alpha $ with $\frac{1}{4} < \alpha < \frac{1}{2}$. These solutions are shown to evolve in $L^2 (\mathbb{R}) \cap L^4 (\mathbb{R})$. This construction is valid, in particular, if the initial data is the characteristic function on an interval of length not an odd multiple of ${\pi / 2}$.