Well-posedness of the Cauchy problem for the fourth-order nonlinear Schrödinger equation

Abstract

The sharp local well-posedness for the one dimensional fourth-order nonlinear Schrödinger equation is established in the Sobolev space $H^s\left(R\right)$ for $s\ge \frac{1}{2}$, which improves the results in Huo and Jia (2007). In addition, we prove that this equation cannot be solved by an iteration scheme based on the Duhamel formula in $H^s\left(R\right)$ for $s<\frac{1}{2}$. Our method relies upon the Bourgain space and a crucial bilinear estimate, which avoids the tedious classification of the location to the highest dispersion modulation.