On the radius of spatial analyticity for defocusing nonlinear Schr\"odinger equations

In this paper we study spatial analyticity of solutions to the defocusing nonlinear Schr"odinger equations $i{u}_t+\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$, given initial data which is analytic with fixed radius. It is shown that the uniform radius of spatial analyticity of solutions at later time $t$ cannot decay faster than $1\mathrm{/}\mathrm{\mid }t\mathrm{\mid }$ as $\mathrm{\mid }t\mathrm{\mid }\to \mathrm{\infty }$. This extends the previous work of Tesfahun for the cubic case $p=3$ to the cases where $p$ is any odd integer greater than $3$.