The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension

Abstract

We show that the Cauchy problem for the quintic NLS on $\mathbf{R}$ is globally well posed in $H^s$ for $4/9<s\leq 1/2$. Since we work below the energy space we cannot immediately use the energy. Instead we use the "I-method" introduced by J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. This method allows us to define a modification of the energy functional that is "almost conserved" and thus can be used to iterate the local result.