Global Well-posedness for the Fourth-order Nonlinear Schrodinger Equation

The local and global well-posedness for the one dimensional fourth-order nonlinear Schrödinger equation are established in the modulation space $M_{2,q}^s$ for $s\ge \frac{1}{2}$ and $2\le q<\mathrm{\infty }$. The local result is based on the $U^p-V^p$ spaces and crucial bilinear estimates. The key ingredient to obtain the global well-posedness is that we achieve a-priori estimates of the solution in modulation spaces by utilizing the power series expansion of the perturbation determinant introduced by Killip-Visan-Zhang for completely integrable PDEs.